Volume Doubling Measures and Heat Kernel Estimates on Self-Similar Sets

Jun Kigami

Author address:

Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan

E-mail address: kigami@i.kyoto-u.ac.jp

To my family, Keita, Rie and Hiroko

Contents

Prologue 1 0.1. Introduction 1 0.2. the Unit square 4

Chapter 1. Scales and Volume Doubling Property of Measures 9 1.1. Scale 9 1.2. Self-similar structures and measures 13 1.3. Volume doubling property 16 1.4. Locally finiteness and gentleness 20 1.5. Rationally ramified self-similar sets 1 23 1.6. Rationally ramified self-similar sets 2 29 1.7. Examples 34

Chapter 2. Construction of Distances 41 2.1. Distances associated with scales 41 2.2. Intersection type 44 2.3. Qdistances adapted to scales 49

Chapter 3. Heat Kernel and Volume Doubling Property of Measures 57 3.1. Dirichlet forms on self-similar sets 57 3.2. Heat kernel estimate 61 3.3. P. c. f. self-similar sets 62 3.4. Sierpinski carpets 68 3.5. Proof of Theorem 3.2.3 72

Appendix 81 A. Existence and continuity of a heat kernel 81 B. Recurrent case and resistance form 84 C. Heat kernel estimate to the volume doubling property 85

Bibliography 87 Assumptions, Conditions and Properties in Parentheses 89 List of Notations 90 Index 91

Abstract

This paper studies the following three problems.

1. When does a measure on a self-similar set have the volume doubling property with respect to a given distance?

2. Is there any distance on a self-similar set under which the contraction mappings have the prescribed values of contractions ratios?

3. When does a heat kernel on a self-similar set associated with a self-similar Dirichlet form satisfy the Li-Yau type sub-Gaussian diagonal estimate? Those three problems turns out to be closely related. We introduce a new class of self-similar set, called rationally ramified self-similar sets containing both the Sierpinski gasket and the (higher dimensional) Sierpinski carpet and give complete solutions of the above three problems for this class. In particular, the volume doubling property is shown to be equivalent to the upper Li-Yau type sub-Gaussian diagonal estimate of a heat kernel.

Received by the editor May 26, 2005; and in revised form September 18, 2006.

2000 Mathematics Subject Classification. Primary 28A80, 60J35; Secondary 31C25, 60J45.

Key words and phrases. self-similar set, volume doubling property, heat kernel.

Prologue

0.1. Introduction

This paper has originated from two naive questions about a self-similar set K. The first one is when a (self-similar) measure μ on K has the volume doubling property ((VD) for short) with respect to a given distance d. Let B_{r}(x, d) = {y|d(x, y) < r} and let V (x, r) be the volume of the ball B_{r}(x, d), i.e. V (x, r) = μ(B_{r}(x, d)). We say that μ has (VD) if and only if

V (x, 2r) ≤ cV (x, r)

for any x and r, where c is independent of x and r. The simplest situation is when V (x, r) = cr^{n} for any r and any x as we can observe in the case of the Lebesgue measures on the Euclidean spaces. Note that in such a case, V (x, r) is homogeneous in space. The next best situation is to have (VD). Under it, we may allow inhomogeneity in space and, at the same time, still have good control of the volume by the distance. (VD) plays an important role in many area of analysis and geometry, for example, harmonic analysis, geometric measure theory, global analysis and so on.

The second question is when a heat kernel p(t, x, y) on a self-similar set satisfies the following type of on-diagonal estimate

(0.1.1)  \frac{c_{1}}{V (x, t^{1/β})} ≤ p(t, x, x) ≤ \frac{c_{2}}{V (x, t^{1/β})},

for t ∈ (0, 1]. The estimate (0.1.1) immediately implies

lim_{t→0} −\frac{log p(t, x, x)}{log t} = lim_{r→0} \frac{1}{β} \frac{log μ(B_{r}(x, d))}{log r}.

This relates the asymptotic behavior of the heat kernel to the multifractal analysis on the measure. (See Falconer [12, 13] about multifractal analysis.) Such a relation has been observed in [20] for post critically finite sets and in [8] for Sierpinski carpets.

Since Barlow and Perkins [9], there have been extensive results on heat kernels on self-similar sets. Mainly those works have focused on sub-Gaussian estimate

(0.1.2)  p(t, x, y) ≈ c_{1}t^{−d_{s}/2} exp ( − c_{2}(\frac{d(x, y)^{β}}{t} )^{1/(β−1)}),

where d_{s} is a positive constant called the spectral dimension, d(·, ·) is a distance and β is a constant with β ≥ 2. This type of estimate has been first established for the “Brownian motion” on the Sierpinski gasket in [9]. Then it has been proven for nested fractals in [33], affine nested fractals in [14] and the Sierpinski carpets in [7]. Note that (0.1.2) gives a homogeneous on-diagonal estimate

(0.1.3)  c_{1}t^{−d_{s}/2} ≤ p(t, x, x) ≤ c_{2}t^{−d_{s}/2}

The homogeneous estimate (0.1.3) is known to require exact match between the measure μ and the form (E, F). To be more precise, let K be the self-similar set associated with a family of contractions {F_{i}}_{i=1,...,N}, i.e., K = ∪_{i=1}^{N}F_{i}(K). We consider heat kernels associated with a self-similar Dirichlet form (E, F), where E is the form and F is the domain of the form, under a self-similar measure μ with weight {μ_{i}}_{i∈S}, where S = {1, …, N}. (E, F) is said to have self-similarity if

E(u, v) = ∑_{i=1}^{N} \frac{1}{r_{i}} E(u∘F_{i}, v∘F_{i})

for any u, v ∈ F, where (r_{1}, …, r_{N}) is a positive vector called resistance scaling ratio. Also a probability measure on K is called a self-similar measure on K with weight {μ_{i}}_{i∈S} if

μ(A) = ∑_{i=1}^{N} μ_{i}μ(F_{i}^{−1}(A))

for any measurable set A. By the results in [20, 28, 8], the homogeneous ondiagonal estimate (0.1.3) holds if the ratio between log r_{i} and log μ_{i} is independent of i. Otherwise, we may only expect inhomogeneous estimate (0.1.1) at the best.

The first and the second questions may look completely independent at a glance. They are, however, closely related. One of the main result in this paper is that the volume doubling property is equivalent to the upper inhomogeneous on-diagonal heat kernel estimate

(0.1.4)  p(t, x, x) ≤ \frac{c}{V (x, t^{1/β})}.

for t ∈ (0, 1]. Moreover, it turns out that the upper estimate (0.1.4) implies the upper and lower estimate (0.1.1). As a consequence, the first and the second questions are virtually the same. In fact, it has been known that (VD) combined with other properties is equivalent to the following Li-Yau type estimate of a heat kernel,

(0.1.5)  p(t, x, y) ≈ \frac{c_{1}}{V (x, t^{1/β})} exp ( − c_{2}(\frac{d(x, y)^{β}}{t} )^{1/(β−1)}).

For example, in the case of Riemannian manifolds, Grigor’yan [16] and Saloff-Coste [38] have shown that (0.1.5) is equivalent to (VD) and the Poincaré inequality. See [19, 17] for other settings. In our case, the self-similarity of the space and the form allow (VD) itself to be equivalent to the heat kernel estimate (0.1.4).

At this point, a careful reader might notice that something is missing. Indeed, we have not mentioned what kind of distance we use in (0.1.1). In the course of our study, the natural distance for a heat kernel estimate like (0.1.1) should be a distance under which the system of contractions {F_{i}}_{i∈S} has an asymptotic contraction ratio {(r_{i}μ_{i})^{α/2}}_{i∈S} for some α, i.e. d(F_{w_{1}...w_{m}}(x), F_{w_{1}...w_{m}}(y)) is asymptotically (γ_{w_{1}} ⋯ γ_{w_{m}})^{α}d(x, y), where γ_{i} = \sqrt{r_{i}μ_{i}} and F_{w_{1}...w_{m}} = F_{w_{1}} ∘ … ∘F_{w_{m}} for w_{1}, …, w_{m} ∈ S. Does such a distance really exist or not? Generalizing this, we have the third question. For a given ratio a = (a_{i})_{i∈S}, is there any distance under which {F_{i}}_{i∈S} has the asymptotic contraction ratio a? A similar problem has been studied in [25] for post critically finite self-similar sets. We will consider broader class of self-similar set with a different approach.

The key idea to study the third question is the notion of a scale, which essentially gives the size of K_{w_{1}...w_{m}} = F_{w_{1}...w_{m}}(K). For a given ratio a = (a_{1}, …, a_{N}), we think of a_{w_{1}...w_{m}} = a_{w_{1}}⋯a_{w_{m}} as the size of K_{w_{1}...w_{m}}. (Note that we do not suppose the existence of any distance at this point. If there were a distance which satisfies d(F_{i}(x), F_{i}(y)) = a_{i}d(x, y), then the size of K_{w_{1}...w_{m}} had to be a_{w_{1}...w_{m}}.) Starting from the scale (i.e. the size of K_{w_{1}...w_{m}}), we will construct a system of fundamental neighborhoods {U_{s}(x)}_{s∈(0,1]}, which is the counterpart of balls with radius s and center x under a distance. See Section 1.3 for details. Now the problem is the existence of a distance whose balls match the virtual balls {U_{s}(x)}, or to be more exact, there is a distance d which satisfies

(0.1.6)  B_{c_{1}s}(x, d) ⊆ U_{s}(x) ⊆ B_{c_{2}s}(x, d)

for any s and any x or not, where c_{1} and c_{2} are independent of s and x. We say that a distance d is adapted to a scale if (0.1.6) holds.

As a whole, we will study three problems in this paper. Introduced in accordance with the appearance in this paper, they are (P1) When does a (self-similar) measure have the volume doubling property with respect to a scale? The volume doubling property with respect to a scale means that

μ(U_{2s}(x)) ≤ cμ(U_{s}(x))

for any s ∈ (0, 1/2] and any x ∈ K. (P2) Is there a good distance which is adapted to a given scale? (P3) When does (0.1.1) hold for the heat kernel associated with a self-similar Dirichlet form and a (self-similar) measure?

(P1), (P2) and (P3) will be studied in Chapter 1, 2 and 3 respectively. Also those three questions are shown to be closely related in the course of discussion. In Chapter 1, we are going to introduce three properties, namely, an elliptic measure (EL), a locally finite scale (LF) and a gentle measure (GE). In short, (VD) turns out to be equivalent to the combination of (EL), (LF) and (GE). See Theorem 1.3.5. In the following sections, we will try to get simpler and effective description of (EL), (LF) and (GE) respectively for a restricted class of self-similar sets called rationally ramified self-similar sets. This class includes post critically finite self-similar sets, the cubes in ℝ^{n} and the (higher dimensional) Sierpinski carpets. Also, for this class, we will give a complete answer to (P2) in Corollary 2.2.8, saying that, for a given ratio a = (a_{1}, …, a_{N}), the scale associated with a satisfies (LF) if and only if there exists a distance which matches to the scale associated with the ratio ((a_{i})^{α})_{i∈S} for some α > 0. Based on those results, close relation between (P1), (P2) and (P3) will be revealed in Chapter 3. In particular, in Theorem 3.2.3, the following three conditions (a), (c) and (d) will be shown to be equivalent for rationally ramified self-similar sets:

(a) μ is (VD) with respect to the scale associated with the ratio (γ_{i})_{i∈S}.

(c) p(t, x, x) ≤ \frac{c}{μ(U\sqrt{t}(x))} for t ∈ (0, 1].

(d) There exist α > 0 and a distance d which is adapted to the scale associated with the ratio ((γ_{i})^{α})_{i∈S} such that (0.1.4) holds, where β = 2/α. Moreover, if any of the above condition is satisfied, then we have full diagonal estimate (0.1.1) and the upper Li-Yau type estimate

(0.1.7)  p(t, x, y) ≤ \frac{c_{1}}{V (x, t^{1/β})} exp ( − c_{2}(\frac{d(x, y)^{β}}{t} )^{1/(β−1)}).

for t ∈ (0, 1]. Combining this results with the conclusion on (P2), we can easily determine self-similar measures for which (0.1.4) holds.

The organization of this paper is as follows. In Section 1.1, we introduce the notion of scales and establish several fundamental facts on this notion. In Section 1.2, we study self-similar structures and self-similar measures under the assumption that K ≠ \overline{V_{0}}. This section gives bases of the discussions in the following sections. Section 1.3 is devoted to showing the equivalence between (VD) and the combination of (EL), (LF) and (GE) as we mentioned above. In Section 1.4, the properties (LF) and (GE) are closely examined. In particular, it is shown that (GE) is a equivalence relation among elliptic scales and (LF) is inherited by the equivalence relation (GE). The notion of rationally ramified self-similar set is introduced in Section 1.5. For this class of self-similar sets, we will find an effective and simple criteria for (LF) and (GE) in Section 1.6. We apply them to examples including post critically self-similar sets and the Sierpinski gasket in Section 1.7. The search of a distance which matches a scale starts at Section 2.1, where we define a pseudodistance associated with a scale. In Section 2.2, the notion of intersection type is introduced to give an answer to the existence problem of a distance adapted to a scale. Using the notion of qdistance, we will simplify the results in the previous two sections in Section 2.3. We will finally encounter with heat kernels in Section 3.1, which is completely devoted to setting up a reasonable framework of self-similar Dirichlet forms and the heat kernel associated with them. In Section 3.2, we establish a theorem to answer (P3), which will be the most important result in this paper. In Sections 3.3 and 3.4, we apply our main theorem to the post critically finite selfsimilar set and the Sierpinski carpets respectively. We need the entire Section 3.5 to complete the main theorem. In Appendixes, we mainly discuss relations between the properties of the heat kernel associated with a local regular Dirichlet from on a general measure-metric space.

0.2. the Unit square

Let us illustrate our main results by applying them to the unit square [0, 1]^{2}, which is naturally self-similar. We denote the square by K and think of it as a subset of ℂ. Namely, K = \{x + y\sqrt{−1}|x, y ∈ [0, 1]\}. The unit square can be regarded as a self-similar set in many ways. First, let f_{1}(z) = z/2, f_{2}(z) = z/2+1/2, f_{3}(z) = z/2 + (1 + \sqrt{−1})/2 and f_{4}(z) = z/2 + \sqrt{−1}/2. Then K = f_{1}(K) ∪ f_{2}(K) ∪ f_{3}(K) ∪ f_{4}(K). According to the terminology in [28], K is the self-similar set with respect to {f_{1}, f_{2}, f_{3}, f_{4}}. K is not post critically finite but, so called, infinitely ramified self-similar set. Roughly speaking if any of f_{i}(K) ∩ f_{j}(K) is not a finite set, then K is called infinitely ramified self-similar set. In this case, K_{1} ∩ K_{2} is a line, where K_{i} = f_{i}(K).

Now let us explain the notion of “rationally ramified” self-similar sets by the unit square, which is the simplest (non trivial) rationally ramified self-similar set. There exists a natural map π from {1, 2, 3, 4}^{ℕ} → K which is defined by π(i_{1}i_{2} …) = ∩_{m≥1}f_{i_{1}...i_{m}}(K). This map π determines the structure of K as a self-similar set. Note that the four line segments in the boundary of K is also self-similar sets. To

1  2^{1}  ^{2}^{4}  ^{3}  ^{1}  ^{2}^{3}^{4} 4  3^{⟷}^{⟷}_{f_{1}(M_{23}) = f_{2}(M_{14})}^{M_{14}}  ^{M_{23}} ^{ϕ(1) = 2}^{ϕ(4) = 3}^{π(2i_{1}i_{2} …) = π(1ϕ(i_{1})ϕ(i_{2}) …)}^{∈ f_{2}(M_{14}) = f_{1}(M_{23}) = K_{1} ∩ K_{2}}^{where i_{1}, i_{2}, … ∈ {1, 4}^{ℕ}}

Figure 0.1. the square as a self-similar set

see this, set M_{14} = \{\sqrt{−1}t|t ∈ [0, 1]\} and M_{23} = \{1 + \sqrt{−1}t|t ∈ [0, 1]\} for example. Then M_{14} = f_{1}(M_{14})∪f_{4}(M_{14}) and M_{23} = f_{2}(M_{23})∪f_{3}(M_{23}) and hence M_{ij} is the self-similar set with respect to {f_{i}, f_{j}}. In other words, M_{ij} = π({i, j}^{ℕ}). Those two self-similar sets M_{14} and M_{23} meet each other at K_{1} ∩ K_{2} under the action of f_{1} and f_{2}. More precisely, let x ∈ K_{1} ∩ K_{2}. Then there exists i_{1}i_{2} … ∈ {1, 4}^{ℕ} such that x = π(2i_{1}i_{2} …) = π(1ϕ(i_{1})ϕ(i_{2}) …), where ϕ : {1, 4} → {2, 3} is defined by ϕ(1) = 2, ϕ(4) = 3. See Figure 0.1.

Note that other intersections K_{i}∩K_{j} have similar descriptions. This is a typical example of rationally ramified self-similar set defined in 1.5, where an intersection of f_{i}(K) ∩ f_{j}(K) itself is a self-similar set and two different expressions (started from i and j respectively) by infinite sequences of symbols can be translated by a simple rewriting rules.

Next, applying the results in Chapter 1, we present the answer to the problem (P1) in this case. In particular, we can determine the class of self-similar measures which have the volume doubling property with respect to the Euclidean distance.

Theorem 0.2.1. A self-similar measure with weight (μ_{1}, μ_{2}, μ_{3}, μ_{4}) has the volume doubling property with respect to the Euclidean distance if and only if μ_{1} = μ_{2} = μ_{3} = μ_{4} = 1/4.

If μ_{i} = 1/4 for all i, then μ is the restriction of the Lebesgue measure on K. So, the situation is very rigid and not quite interesting. In general, however, we can find richer structure of the volume doubling (self-similar) measures (even in the case of unit square). To see this, we are going to change the self-similar structure of the unit square.

From now on, K is regarded as a self-similar set with respect to nine contractions {F_{i}}_{i=1,...,9} in stead of four contractions {f_{i}}_{i=1,...,4} as above. Set p_{1} = 0, p_{2} = 1/2, p_{3} = 1, p_{4} = 1+\sqrt{−1}/2, p_{5} = 1+\sqrt{−1}, p_{6} = 1/2+\sqrt{−1}, p_{7} = \sqrt{−1}, p_{8} = \sqrt{−1}/2

a  b  a c  d  c a  b  a  ^{a = a_{1} = a_{3} = a_{5} = a_{7}}^{b = a_{2} = a_{6}}^{c = a_{4} = a_{8}}^{d = a_{9}}

Figure 0.2. Weakly symmetric ratio

and p_{9} = 1/2 + \sqrt{−1}/2. Define F_{i}(z) = (z − p_{i})/3 + p_{i} for i = 1, …, 9. Then the square K is the self-similar set with respect to {F_{i}}_{i∈S}, where S = {1, …, 9}, i.e. K = ∪_{i∈S}F_{i}(K). In this case, we also have the natural map π from S^{ℕ} = {w_{1}w_{2} … |w_{i} ∈ S} to K defined by π(w_{1}w_{2} …) = ∩_{m≥0}F_{w_{1}...w_{m}}(K). Examining the intersection of F_{1}(K) and F_{2}(K), one may notice that x = π(1w_{1}w_{2} …) = π(2φ(w_{1})φ(w_{2}) …) for any x ∈ F_{1}(K) ∩ F_{2}(K), where w_{1}w_{2} … ∈ {3, 4, 5}^{ℕ} and φ : {3, 4, 5} → {1, 8, 7} is given by φ(3) = 1, φ(4) = 8 and φ(5) = 7. Also for any y ∈ F_{1}(K) ∩ F_{8}(K), we have y = π(8v_{1}v_{2} …) = π(1ψ(v_{1})ψ(v_{2}) …), where v_{1}v_{2} … ∈ {1, 2, 3}^{ℕ} and ψ : {1, 2, 3} → {7, 6, 5} is given by ψ(1) = 7, ψ(2) = 6 and ψ(3) = 5. This is again a typical example of a rationally ramified self-similar set.

Under this self-similar structure, self-similar volume doubling measures are much richer than before. The following condition will play an important role to solve all the three problems (P1), (P2) and (P3).

Definition 0.2.2. A ratio (a_{i})_{i∈S} ∈ (0, 1)^{S} is called weakly symmetric if and only if a_{i} = a_{φ(i)} for any i ∈ {3, 4, 5} and a_{j} = a_{ψ(j)} for any j ∈ {1, 2, 3}.

Note that a ratio (a_{i})_{i∈S} is weakly symmetric if and only if

a_{1} = a_{3} = a_{5} = a_{7}, a_{2} = a_{6} and a_{4} = a_{8},

See Figure 0.2. First our results on (P1) in Chapter 1 yields the following characterization of the class of self-similar measures which are volume doubling with respect to the Euclidean distance.

Theorem 0.2.3. A self-similar measure with weight (μ_{i})_{i∈S} has the volume doubling property with respect to the Euclidean distance if and only if (μ_{i})_{i∈S} is weakly symmetric.

As we have explained in the introduction, the main result of this paper is roughly the equivalence of the three properties: the volume doubling property of a measure, the existence of “asymptotically self-similar” distance and the upper and lower on-diagonal heat kernel estimate (0.1.1). In accordance with this spirit, being weakly symmetric gives an answer to (P2) as well. More precisely, the results in Chapter 2 gives the following theorem.

Theorem 0.2.4. Let (a_{i})_{i∈S} ∈ (0, 1)^{S}. (a_{i})_{i∈S} is weakly symmetric if and only if there exists a distance d which is adapted to the scale associated with a ratio ((a_{i})^{α})_{i∈S} for some α > 0.

Naturally, weakly symmetric ratios appear again in our result on the problem (P3). To consider heat kernels, we regard K as a subset of ℝ^{2} in the natural manner. Let ν be the restriction of the Lebesgue measure and let F = W^{1,2}(K). W^{1,2}(K) is the Sobolev space defined by

W^{1,2}(K) = \{f|f ∈ L^{2}(K, ν), \frac{∂f}{∂x} , \frac{∂f}{∂y} ∈ L^{2}(K, ν)\},

where \frac{∂f}{∂x} and \frac{∂f}{∂y} are the partial derivatives in the sense of distribution. Note that ν is the self-similar measure with weight (1/9, …, 1/9). For any f, g ∈ F, set

E(f, g) = ∫_{K} (\frac{∂f}{∂x} \frac{∂g}{∂x} + \frac{∂f}{∂y} \frac{∂g}{∂y} )dxdy.

Then (E, F) is a local regular Dirichlet form on L^{2}(K, ν) and the corresponding diffusion process is the Brownian motion which is reflected at the boundary of K. Moreover, the associated heat kernel satisfy the Gaussian type estimate

p(t, x, y) ≈ \frac{c_{1}}{t} exp ( − c_{2} \frac{|x − y|^{2}}{t} )

for t ∈ (0, 1]. This form (E, F) has the self-similarity with the resistance scaling ratio (1, …, 1), i.e.

E(f, g) = ∑_{i∈S} E(f ∘ F_{i}, g ∘ F_{i}).

for any f, g ∈ F. Let μ be a self-similar measure with weight (μ_{i})_{i∈S}. Then by [8], making slight modifications, we may regard (E, F) as a local regular Dirichlet form on L^{2}(K, μ). At this time, the corresponding diffusion process is the time change of the Brownian motion. Let p_{μ}(t, x, y) be the associated heat kernel. (The heat kernel does exists and is jointly continuous in this case.) Then the results on (P3) implies the following.

Theorem 0.2.5. There exist α ∈ (0, 1] and a distance d such that d is adapted to the scale associated with the ratio ((μ_{i})^{α/2})_{i∈S} and the upper Li-Yau estimate (0.1.7) for p_{μ}(t, x, y) holds with β = 2/α, if and only if (\sqrt{μ_{i}})_{i∈S} is weakly symmetric. Moreover, either of the above conditions suffices the upper and the lower on-diagonal estimate (0.1.1) for p_{μ}(t, x, y).

CHAPTER 1 Scales and Volume Doubling Property of Measures

1.1. Scale

In this section, we introduce a notion of scales. A scale gives a fundamental system of neighborhoods of the shift space, which is the collection of infinite sequences of finite symbols. Later in Section 1.3, we will define a family of “balls” of a self-similar set through a scale.

Notation. For a set V, we define ℓ(V) = {f|f : V → ℝ}. If V is a finite set, ℓ(V) is considered to be equipped with the standard inner product (·, ·)_{V} defined by (u, v)_{V} = ∑_{p∈V} u(p)v(p) for any u, v ∈ ℓ(V). Also |u|_{V} = \sqrt{(u, u)_{V}} for any u ∈ ℓ(V).

Now we define basic notions on the word spaces and the shift space. Let S be a finite set.

Definition 1.1.1. (1) For m ≥ 0, the word space of length m, W_{m}(S), is defined by

W_{m}(S) = S^{m} = {w|w = w_{1} … w_{m}, w_{i} ∈ S for any i = 1, …, m}.

In particular W_{0}(S) = {∅}, where ∅ is called the empty word. Also W_{∗}(S) = ∪_{m≥0}W_{m}(S) and W_{#}(S) = ∪_{m≥1}W_{m}(S). For w ∈ W_{m}(S), we define |w| = m and call it the length of the word w.

(2) For w, v ∈ W_{∗}(S), we define wv ∈ W_{∗}(S) by wv = w_{1} … w_{m}v_{1} … v_{n}, where w = w_{1} … w_{m} and v = v_{1} … v_{n}. Also for w^{1}, w^{2} ∈ W_{∗}(S), we write w^{1} ≤ w^{2} if and only if w^{1} = w^{2}v for some v ∈ W_{∗}(S).

(3) The (one sided) shift space Σ(S) is defined by

Σ(S) = S^{ℕ} = {ω|ω = ω_{1}ω_{2} …, ω_{i} ∈ S for any i ≥ 1}.

The shift map σ : Σ(S) → Σ(S) is defined by σ(ω_{1}ω_{2} …) = ω_{2}ω_{3} . . .. For each i ∈ S, we define σ_{i} : Σ(S) → Σ(S) by σ_{i}(ω) = iω_{1}ω_{2} …, where ω = ω_{1}ω_{2} . . .. For w = w_{1} … w_{m} ∈ W_{∗}(S), σ_{w} = σ_{w_{1}} ∘ … ∘ σ_{w_{m}} and Σ_{w}(S) = σ_{w}(Σ(S)).

(4) The extended shift map σ : W_{∗}(S) → W_{∗}(S) is defined by σ(∅) = ∅ and σ(w_{1} … w_{m}) = w_{2} … w_{m} for any w ∈ W_{#}. Also we extend σ_{i} : W_{∗}(S) → W_{∗}(S) by σ_{i}(w_{1} … w_{m}) = iw_{1} … w_{m}.

Note that ≤ is a partial order of W_{∗}(S). We write w^{1} < w^{2} if and only if w^{1} ≤ w^{2} and w^{1} ≠ w^{2}. If no confusion can occur, we omit S in the above notations. For example, we write W_{m} in stead of W_{m}(S).

The shift space Σ has a product topology as an infinite product of a finite set S. Under this topology, Σ is compact and metrizable. See [28] for details.

Definition 1.1.2. (1) Let Λ ⊂ W_{∗} be a finite set. Λ is called a partition of Σ if and only if Σ = ∪_{w∈Λ}Σ_{w} and Σ_{w} ∩ Σ_{v} = ∅ for any w ≠ v ∈ Λ.

(2) Let Λ_{1} and Λ_{2} be partitions of Σ. Λ_{1} is said to be a refinement of Λ_{2} if and only for any w^{1} ∈ Λ_{1}, there exists w^{2} ∈ Λ_{2} such that w^{1} ≤ w^{2}. We write Λ_{1} ≤ Λ_{2} if Λ_{1} is a refinement of Λ_{2}.

For a partition Λ, {Σ_{w}}_{w∈Λ} is a division of Σ and may be thought of as an approximation of Σ. Note that “≤” is a partial order of the collection of partitions. If Λ_{1} ≤ Λ_{2}, then {Σ_{w}}_{w∈Λ_{1}} contains finer structure of Σ than {Σ_{w}}_{w∈Λ_{2}}.

Next we introduce the notion of a scale, which is a monotonically decreasing family of partitions.

Definition 1.1.3 (Scales). A family of partitions of Σ, {Λ_{s}}_{0<s≤1}, is called a scale on Σ if and only if it satisfies (S1) and (S2): (S1) Λ_{1} = W_{0}. Λ_{s_{1}} ≤ Λ_{s_{2}} for any 0 < s_{1} ≤ s_{2} ≤ 1. (S2) min{|w||w ∈ Λ_{s}} → +∞ as s ↓ 0.

Let {Λ_{s}}_{0<s≤1} be a scale on Σ. For any ω ∈ Σ and any s ∈ (0, 1], choose w ∈ Λ_{s} so that ω ∈ Σ_{w}, (such a w uniquely exists), and set U_{s}(ω) = Σ_{w}. Then {U_{s}(ω)}_{s∈(0,1]} is a system of fundamental neighborhoods of ω. We will think of U_{s}(ω) as a “ball” with radius s and center ω even if there may not be a corresponding distance.

In the rest of this section, we will try to understand the basics on scales. First problem is how to describe the structure of a scale.

Definition 1.1.4. Let 𝒮 = {Λ_{s}}_{0<s≤1} be a scale on Σ. For w ∈ W_{∗}. We define

R_{w}(𝒮) = {s|s ∈ (0, 1], there exists w′ ∈ W_{∗} such that w < w′ and w′ ∈ Λ_{s}},
C_{w}(𝒮) = {s|s ∈ (0, 1], w ∈ Λ_{s}},
L_{w}(𝒮) = {s|s ∈ (0, 1], there exists w′ ∈ W_{∗} such that w′ < w and w′ ∈ Λ_{s}}.

For ease of notation, we use R_{w}, C_{w} and L_{w} instead of R_{w}(𝒮), C_{w}(𝒮) and L_{w}(𝒮) if no confusion can occur. Note that R_{∅} = ∅ and that C_{∅} contains 1.

Lemma 1.1.5. Let 𝒮 = {Λ_{s}}_{0<s≤1} be a scale on Σ. For w ∈ W_{∗}.

(1) There exist r : W_{#} → (0, 1] and l : W_{∗} → (0, 1] such that, for any w, l(w) ≤ r(w) and R_{w} ⊇ (r(w), 1], C_{w} ⊇ (l(w), r(w)) and L_{w} ⊇ (0, l(w)).

(2) For any w ∈ W_{∗} and any i ∈ S, C_{wi} ∪ L_{wi} = L_{w}. In particular, r(wi) = l(w) and l(wi) ≤ l(w).

(3) max{l(w)|w ∈ W_{m}} → 0 as m → ∞.

Proof. (1) Since 1 ∈ L_{∅}, R_{w} ≠ ∅ for w ∈ W_{#}. Also by (S2), L_{w} ≠ ∅ for any w ∈ W_{∗}. Using (S1), we see that x < y for any x ∈ L_{w} ∪ C_{w} and any y ∈ R_{w}. Therefore the Dedekind theorem implies that there exists r(w) such that (0, r(w)) ⊆ L_{w} ∪ C_{w} and (r_{(}w), 1] ⊆ R_{w}. In the same manner, we have l(w).

(2) Note that s ∈ L_{wi} ∪ C_{wi} if and only if there exists w′ ∈ W_{∗} such that w′ ≤ wi and w′ ∈ Λ_{s}. This immediately implies that L_{wi} ∪ C_{wi} ⊆ L_{w}. Suppose s ∈ L_{w}. There exists w′ ∈ Λ_{s} such that wii … ∈ Σ_{w′}. Since w′′ ∉ Λ_{s} if w ≤ w′′, it follows that w′ ≤ wi. Therefore s ∈ L_{wi} ∪ C_{wi}. The rest of the statement is obvious.

(3) Let a_{m} = max{l(w)|w ∈ W_{m}}. Then a_{m} ≥ a_{m+1} for any m ≥ 1. Set α = lim_{m→∞} a_{m}. Suppose α > 0. Choosing w^{m} ∈ W_{m} so that l(w^{m}) ≥ α, we see that Λ_{α/2} contains w′ ≤ w^{m} for any m ≥ 1. Therefore, Λ_{α/2} is an infinite set. This contradiction implies that α = 0. □

In general, R_{w} can be either (r(w), 1] or [r(w), 1]. To remove this ambiguity, we introduce the notion of a right continuous scale.

Definition 1.1.6. A scale {Λ_{s}}_{0<s≤1} on Σ is called right continuous if and only if R_{w} = [r(w), 1] for any w ∈ W_{∗}.

Lemma 1.1.5 implies that if {Λ_{s}}_{0<s≤1} is right continuous then L_{w} = (0, l(w)) and C_{w} = [l(w), r(w)).

Proposition 1.1.7. A scale {Λ_{s}}_{0<s≤1} on Σ is right continuous if and only if, for any s, there exists ϵ > 0 such that Λ_{s′} = Λ_{s} for any s′ ∈ [s, s + ϵ).

Right continuous scales are completely determined by l : W_{∗} → (0, 1], which will be called the gauge function of the scale. See Theorem 1.1.10 for details.

Definition 1.1.8. A function g : W_{∗} → (0, 1] is called a gauge function on W_{∗} if it satisfies (G1) and (G2): (G1) g(wi) ≤ g(w) for any w ∈ W_{∗} and any i ∈ S. (G2) max{g(w)|w ∈ W_{m}} → 0 as m → ∞.

The following proposition is immediate by Lemma 1.1.5.

Proposition 1.1.9. Let 𝒮 be a scale on Σ. Then the function l : W_{∗} → (0, 1] defined in Lemma 1.1.5 is a gauge function on W_{∗}. We call l the gauge function of the scale 𝒮.

Naturally there exists a one to one correspondence between the (right continuous) scales and the gauge functions.

Theorem 1.1.10. Let g be a gauge function on W_{∗}. Define Λ_{s}(g) by

(1.1.8)  Λ_{s}(g) = {w|w = w_{1} … w_{m} ∈ W_{∗}, g(w_{1} … w_{m−1}) > s ≥ g(w)}

for any s ∈ (0, 1]. (We regard g(w_{1} … w_{m−1}) as 2 for w = ∅.) Then {Λ_{s}(g)}_{0<s≤1} is a right continuous scale on Σ. {Λ_{s}(g)}_{0<s≤1} is called the scale induced by the gauge function g. Conversely, let 𝒮 = {Λ_{s}}_{0<s≤1} be a right continuous scale on Σ and let l be its gauge function. Then the scale induced by the gauge function l coincides with 𝒮.

Proof. We write Λ_{s} = Λ_{s}(g) for ease of notation. First we show that Λ_{s} is a finite set for any s. By (G2), there exists m ≥ 1 such that s ≥ g(w) for any w ∈ W_{m}. Now if g(v_{1} … v_{n−1}) > s ≥ g(v_{1} … v_{n}), then n ≤ m. Therefore Λ_{s} ⊂ ∪_{m=0}^{m}W_{m}. Hence Λ_{s} is a finite set.

Next we show that Λ_{s} is a partition. Let ω = ω_{1}ω_{2} … ∈ Σ. (G2) implies that g(ω_{1} … ω_{m}) → 0 as m → ∞. Hence there exists a unique m such that g(ω_{1} … ω_{m−1}) > s ≥ g(ω_{1} … ω_{m}). Therefore ∪_{w∈Λ_{s}}Σ_{w} = Σ. Also the uniqueness of m implies that Σ_{w^{1}} ∩ Σ_{w^{2}} = ∅ if w^{1} ≠ w^{2} ∈ Λ_{s}. So Λ_{s} is a partition.

To show (S1), since g(∅) ≤ 1, we have Λ_{1} = W_{0}. Let s_{1} < s_{2} and let w = w_{1} … w_{m} ∈ Λ_{s_{1}}. Then g(w_{1} … w_{k−1}) > s_{2} ≥ g(w_{1} … w_{k}) for some k ≤ m. This implies that w_{1} … w_{k} ∈ Λ_{s_{2}}. Therefore Λ_{s_{1}} ≤ Λ_{s_{2}}.

If s < min_{w∈W_{m}} g(w), then min{|w||w ∈ Λ_{s}} ≥ m. This shows (S2).

Since R_{w} = [g(w_{1} … w_{m−1}), 1] for w = w_{1} … w_{m}, {Λ_{s}}_{0<s≤1} is right continuous.

Finally, let l be the gauge function of a scale 𝒮 = {Λ_{s}}_{0<s≤1}. Then Λ_{s} = {w|s ∈ C_{w}} = Λ_{s}(l). This completes the proof of the theorem. □

Hereafter, we only consider right continuous scales.

Definition 1.1.11. A right continuous scale 𝒮 = {Λ_{s}}_{0<s≤1} on Σ is called elliptic if and only if it satisfies the following two conditions (EL1) and (EL2): (EL1) Λ_{s} ∩ Λ_{α_{1}s} = ∅ for any s ∈ (0, 1], where α_{1} is independent of s. (EL2) There exist α_{2} ∈ (0, 1) and n ≥ 1 such that

max{|v||v ∈ W_{∗}, wv ∈ Λ_{α_{2}s}} ≤ n

for any s ∈ (0, 1] and any w ∈ Λ_{s}.

Roughly speaking, a scale is elliptic if the differences between Λ_{s} and Λ_{αs} are uniform with respect to s. This become clearer when we describe (EL1) and (EL2) in terms of gauge functions.

Proposition 1.1.12. Let 𝒮 = {Λ_{s}}_{0<s≤1} be a right continuous scale on Σ and let l be its gauge function.

(1) 𝒮 satisfies (EL1) if and only if there exists β_{1} ∈ (0, 1) such that l(wi) ≥ β_{1}l(w) for any w ∈ W_{∗} and any i ∈ S.

(2) 𝒮 satisfies (EL2) if and only if there exist β_{2} ∈ (0, 1) and n ≥ 1 such that l(wv) ≤ β_{2}l(w) for any w ∈ W_{∗} and any v ∈ W_{n}.

Proof. (1) First suppose for any β_{1} ∈ (0, 1) there exist w ∈ W_{∗} and i ∈ S such that l(wi) < β_{1}l(w). In particular, we assume that β_{1} < 1/2. Note that wi ∈ Λ_{s} for s ∈ [l(wi), l(w)). If s_{1} = l(w)/2, then Λ_{s_{1}} ∩ Λ_{αs_{1}} contains wi for any α ∈ [2β_{1}, 1]. Hence 𝒮 does not satisfy (EL1).

Conversely, assume that there exists β_{1} ∈ (0, 1) such that l(wi) ≥ β_{1}l(w) for any w ∈ W_{∗} and any i ∈ S. Let w = w_{1} … w_{m} ∈ Λ_{s}. Then l(w_{1} … w_{m−1}) > t ≥ l(w) ≥ β_{1}l(w_{1} … w_{m−1}). Therefore l(w) > β_{1}s. This implies that w ∉ Λ_{β_{1}s}. Hence Λ_{s} ∩ Λ_{β_{1}s} is empty for any s ∈ (0, 1].

(2) Assume that there exist β_{2} ∈ (0, 1) and n ≥ 1 such that l(wv) ≤ β_{2}l(w) for any w ∈ W_{∗} and any v ∈ W_{n}. If w ∈ Λ_{s}, then s ≥ l(w). Therefore, β_{2}s ≥ β_{2}l(w) ≥ l(wv) for any v ∈ W_{n}. Hence if wv′ ∈ Λ_{β_{2}s}, then |v|′ ≤ n. Thus we obtain (EL2) with α_{2} = β_{2}.

Conversely, suppose that, for any β ∈ (0, 1) and any k ≥ 1, there exist w ∈ W_{∗} and v ∈ W_{k} such that βl(w) ≤ l(wv). Let s = l(w). Here, if necessary, replacing w = w_{1} … w_{i} by w = w_{1} … w_{j} for some 0 ≤ j ≤ i, we may assume that w ∈ Λ_{s}. (Then, in general, |v| ≥ k.) Now choose β > α_{2} and k ≥ n. Then α_{2}s < βs = βl(w) ≤ l(wv). Therefore there exists v′ ∈ W_{∗} such that wvv′ ∈ α_{2}s. By (EL2), |vv′| ≤ n. This contradicts to the fact that |v| ≥ k. □

The following fact will be used later in many places.

Lemma 1.1.13. Let 𝒮 = {Λ_{s}}_{0<s≤1} be a scale on Σ satisfying (EL1) and let l be its gauge function on W_{∗}. Then there exists a constant c > 0 such that l(w) ≤ s ≤ cl(w) for any s ∈ (0, 1] and any w ∈ Λ_{s}.

Proof. Let w = w_{1} … w_{m} ∈ Λ_{s}. Then l(w_{1} … w_{m−1}) > s ≥ l(w). By Proposition 1.1.12-(1), l(w) ≥ β_{1}l(w_{1} … w_{m−1}). Therefore, if c = 1/β_{1}, then cl(w) ≥ s ≥ l(w). □

Next we define a multiplication of two scales and a power of a scale.

Definition 1.1.14. (1) For i = 1, 2, let 𝒮_{j} be a scale on Σ and let l_{j} be its gauge function. Then we use 𝒮_{1}·𝒮_{2} to denote the sale induced by the gauge function l_{1}l_{2}.

(2) Let 𝒮 be a scale on Σ. Then for α > 0, the scale induced by the gauge function l^{α} is denoted by 𝒮^{α}.

If 𝒮 = {Λ_{s}}_{0<s≤1}, then 𝒮^{α} = {Λ_{s^{1/α}}}_{0<s≤1}.

Lemma 1.1.15. (1) If 𝒮_{1} and 𝒮_{2} are elliptic scales on Σ, then 𝒮_{1} · 𝒮_{2} is elliptic.

(2) Let 𝒮 be a scale on Σ and let α > 0. Then 𝒮 is elliptic if and only if 𝒮^{α} is elliptic.

Finally we introduce an important class of scales.

Definition 1.1.16. Let a = (a_{i})_{i∈S} ∈ (0, 1)^{S}. Define g_{a} : W_{∗} → (0, 1] by g_{a}(w) = a_{w} = a_{w_{1}}a_{w_{2}} … a_{w_{m}} for w = w_{1} … w_{m} ∈ W_{∗}. g_{a} is called the self-similar gauge function on W_{∗} with weight a. Also the scale induced by g_{a} is called the selfsimilar scale with weight a and is denoted by 𝒮(a). We also write Λ_{s}(g_{a}) = Λ_{s}(a). We use 𝔖(Σ) to denote the collection of self-similar scales on Σ.

We often identify 𝔖(Σ) with (0, 1)^{S} through the natural correspondence a → 𝒮(a). Note that a self-similar scale is elliptic.

1.2. Self-similar structures and measures

The notion of self-similar structure is a purely topological formulation of selfsimilar sets.

Definition 1.2.1. (1) Let K be a compact metrizable topological space and let S be a finite set. Also, let F_{i}, for i ∈ S, be a continuous injection from K to itself. Then, (K, S, {F_{i}}_{i∈S}) is called a self-similar structure if there exists a continuous surjection π : Σ → K such that F_{i} ∘ π = π ∘ σ_{i} for every i ∈ S.

(2) Let L = (K, S, {F_{i}}_{i∈S}) be a self-similar structure. Define the critical set C_{L} and the post critical set P_{L} by C_{L} = π^{−1}(∪_{i≠j∈S}(F_{i}(K) ∩ F_{j}(K))) and P_{L} = ∪_{n≥1}σ^{n}(C_{L}). Also define V_{0} = π(P_{L}).

(3) A self-similar structure L = (K, S, {F_{i}}_{i∈S}) is said to be strongly finite if and only if sup_{x∈K} #(π^{−1}(x)) < +∞, where #(A) is the number of elements of a set A.

Notation. Let L = (K, S, {F_{i}}_{i∈S}) be a self-similar structure. For w = w_{1} … w_{m} ∈ W_{∗}, we define F_{w} = F_{w_{1}} ∘ … ∘ F_{w_{m}} and K_{w} = F_{w}(K). In particular, if w = ∅ ∈ W_{0}, then F_{w} is thought of as the identity map of K and K_{w} = K.

If (K, S, {F_{i}}_{i∈S}) is a self-similar structure, then

K = ⋃_{i∈S} F_{i}(K).

In other words, K is the self-similar set with respect to maps {F_{i}}_{i∈S}. The set V_{0} is a kind of “boundary” of K. Indeed, for any w, v ∈ W_{∗} with Σ_{w} ∩ Σ_{v} = ∅, K_{w} ∩ K_{v} = F_{w}(V_{0}) ∩ F_{v}(V_{0}). Moreover, V_{0} = ∅ if and only if π is bijective and K is identified with Σ. V_{0} is thought of as a characteristic of “complexity” of the self-similar structure.

Throughout this section, we fix a self-similar structure L = (K, S, {F_{i}}_{i∈S}).

Theorem 1.2.2. K ≠ \overline{V}_{0} if and only if int(\overline{V}_{0}) = ∅.

Proof. Assume that int(\overline{V}_{0}) ≠ ∅. Then \overline{V}_{0} ⊇ K_{w} for some w ∈ W_{∗}. Let |w| = m. Let x ∈ K_{w}. If x ∈ K_{w} ∩ (∪_{v∈W_{m},v≠w}K_{v}), then x ∈ F_{w}(V_{0}) by [28, Proposition 1.3.5]. Hence (F_{w})^{−1}(x) ∈ \overline{V}_{0}. Next we assume that x ∉ K_{w} ∩ (∪_{v∈W_{m},v≠w}K_{v}). This assumption is equivalent to that π^{−1}(x) ⊂ Σ_{w}. Since x ∈ \overline{V}_{0}, there exist ω(1), ω(2), … ∈ P such that π(ω(i)) → x as i → ∞. Choosing a subsequence, we may assume that there exists ω ∈ Σ such that ω(i) → ω as i → ∞. The continuity of π implies that π(ω) = x. Hence ω ∈ Σ_{w}. Now σ^{m}ω(i) → σ^{m}ω as i → ∞ and (F_{w})^{−1}(x) = π(σ^{m}ω). Since σ^{m}ω(i) ∈ P, it follows that (F_{w})^{−1}(x) ∈ \overline{V}_{0}. Thus we see that (F_{w})^{−1}(x) ∈ \overline{V}_{0} for any x ∈ K_{w}. This immediately implies that \overline{V}_{0} = K. The converse is obvious. □

Next we introduce a class of non-degenerate measures on a self-similar structure.

Definition 1.2.3. M(K) is defined to be the collection of Borel regular measures on K satisfying the following conditions (M1), (M2) and (M3): (M1) μ is a finite Borel regular measure on K. (M2) For any w ∈ W_{∗}, μ(K_{w}) > 0 and μ(F_{w}(V_{0})) = 0. (M3) μ({x}) = 0 for any x ∈ K. Also M_{1}(K) = {μ|μ ∈ M(K), μ(K) = 1}.

Theorem 1.2.4. Assume that K ≠ \overline{V}_{0}. Let μ be a finite Borel regular measure on K with μ(K) > 0. Then μ ∈ M(K) if the following condition (ELm) holds: (ELm) there exists γ > 0 such that μ(K_{wi}) ≥ γμ(K_{w}) for any w ∈ W_{∗} and any i ∈ S.

Definition 1.2.5. A finite Borel regular measure μ on K is called an elliptic measure if and only if it satisfies (ELm).

Remark. In [28, Section 3.4], a Borel regular measure μ satisfying (ELm) is called a γ-elliptic measure. By the proof of [28, Lemma 3.4.1], it follows that if μ is elliptic then there exists δ ∈ (0, 1) and m ≥ 1 such that μ(K_{wv}) ≤ δμ(K_{w}) for any w ∈ W_{∗} and any v ∈ W_{m}.

Proof. (M1) is immediate. Since K ≠ \overline{V}_{0}, there exist k ∈ ℕ and v ∈ W_{k} such that K_{v} ∩ \overline{V}_{0} = ∅. Since μ is a finite Borel regular measure, for any ϵ > 0, we find an open set O which satisfies F_{w}(V_{0}) ⊂ O and μ(O) ≤ μ(F_{w}(V_{0})) + ϵ. Set Q = {τ|τ ∈ W_{∗}, K_{τ} ⊂ O |τ| ≥ |w|}. As O is open, O = ∪_{τ∈Q}K_{τ}. Define Q_{∗} = {τ|τ ∈ Q, there exists no τ′ ∈ Q such that τ < τ′}. Then {K_{τv}}_{τ∈Q_{∗}} is mutually disjoint. Also, for any τ ∈ Q_{∗}, since K_{τv} ∩ V_{|w|} = ∅, we see that F_{w}(V_{0}) ∩ K_{τv} = ∅. Therefore, by the fact that μ is elliptic,

μ(F_{w}(V_{0})) ≤ μ(O\ ∪_{τ∈Q_{∗}} K_{τv}) = μ(O) − ∑_{τ∈Q_{∗}} μ(K_{τv})

≤ μ(O) − γ^{k} ∑_{τ∈Q_{∗}} μ(K_{τ}) ≤ (1 − γ^{k})μ(O) ≤ (1 − γ^{k})(μ(F_{w}(V_{0})) + ϵ)

Since this holds for any ϵ > 0, it follows that μ(F_{w}(V_{0})) = 0. Thus we obtain (M2).

To show (M3), let x = π(ω) for ω ∈ Σ. By the above remark, we see that μ(K_{ω_{1}...ω_{mn}}) ≤ δ^{n} for any n. Therefore, μ({x}) = μ(∩_{n≥1}K_{ω_{1}...ω_{mn}}) = 0. □

A immediate example of a elliptic measure is a self-similar measure.

Definition 1.2.6. Let (μ_{i})_{i∈S} ∈ (0, 1)^{S} satisfy ∑_{i∈S} μ_{i} = 1. A Borel regular probability measure μ is called a self-similar measure with weight (μ_{i})_{i∈S} if and only if

(1.2.1)  μ(A) = ∑_{i∈S} μ_{i}μ((F_{i})^{−1}(A))

for any Borel set A.

It is known that, for any weight (μ_{i})_{i∈S}, there exists a unique Borel regular probability measure on k that satisfies (1.2.1). See [28, Section 1.4]. In our case, we have the following theorem.

Theorem 1.2.7. Assume that K ≠ \overline{V}_{0}. Let μ be a self-similar measure with weight (μ_{i})_{i∈S} ∈ (0, 1)^{S} with ∑_{i∈S} μ_{i} = 1. Then, μ(K_{w}) = μ_{w} for any w ∈ W_{∗}, where μ_{w} = μ_{w_{1}} ⋯ μ_{w_{m}} for w = w_{1} … w_{m}. In particular, μ is elliptic and μ ∈ M_{1}(K).

Proof. Let O = K\\overline{V}_{0}. For any w ∈ W_{m}, K_{w} = F_{w}(O) ∪ F_{w}(\overline{V}_{0}) and F_{w}(O) ∩ F_{w}(\overline{V}_{0}) = ∅. By [28, Proposition 1.3.5], F_{w}(O) ∩ K_{v} = ∅ for any v ∈ W_{m} with w ≠ v. Therefore, F_{w}(O) is open. Moreover, since V_{0} ⊆ V_{m} = ∪_{v∈W_{m}}F_{w}(V_{0}), it follows that F_{w}(O) ∩ V_{0} = ∅. This implies that F_{w}(O) ∩ \overline{V}_{0} = ∅. Hence F_{w}(O) ⊆ O.

On the other hand, by (1.2.1),

μ(F_{w}(O)) = ∪_{v∈W_{m}}μ_{v}μ((F_{v})^{−1}F_{w}(O)) = μ_{w}μ(O).

Therefore, if O_{m} = ∪_{w∈W_{m}}F_{w}(O), we obtain μ(O_{m}) = ∑_{w∈W_{m}} μ(F_{w}(O)) = μ(O). Note that O_{m} ⊆ O. For sufficiently large m, there exists w ∈ W_{m} such that K_{w} ⊂ O. Since F_{w}(\overline{V}_{0}) ∩ O_{m} = ∅, F_{w}(\overline{V}_{0}) ∈ O\O_{m}. Therefore μ(F_{w}(\overline{V}_{0})) = 0. By (1.2.1), μ(\overline{V}_{0}) = 0 and therefore μ(O) = μ(O_{k}) = 0 for any k ≥ 1. This implies that μ(∪_{w∈W_{k}}F_{w}(\overline{V}_{0})) = 0 for any k ≥ 1. Hence for any w ∈ W_{∗}, μ(K_{w}) = μ(F_{w}(O)) = μ_{w}μ(O) = μ_{w}. This immediately shows that μ is elliptic. Now by Theorem 1.2.4, we verify μ ∈ M_{1}(K). □

Remark. In the above proof, it was shown that if K ≠ \overline{V}_{0}, then L satisfy an “intrinsic” open set condition: there exists a nonempty intrinsic open subset O ⊂ K (i.e. O is open with respect to the topology of K) such that F_{i}(O) ⊆ O and F_{i}(O) ∩ F_{j}(O) = ∅ for any i ≠ j ∈ S.

Remark. We conjecture that the converse of the above theorem is true: if every self-similar measure μ belongs to M_{1}(K) (and hence μ(K_{w}) = μ_{w} for any w ∈ W_{∗}, where (μ_{i})_{i∈S} is the weight of μ), then K ≠ \overline{V}_{0}.

We may define a natural gauge function associated with a measure as follows.

Proposition 1.2.8. Let μ ∈ M^{1}(K). Define g_{μ} : W_{∗} → (0, 1] by g_{μ}(w) = μ(K_{w}). Then g_{μ} is a gauge function on W_{∗}.

Proof. (G1) is immediate. To prove (G2), assume that there exists α > 0 such that max{μ(K_{w})|w ∈ W_{m}} ≥ 2α for any m ≥ 1. Then A = {w|w ∈ W_{∗}, μ(K_{w}) ≥ α} is an infinite set. Let T = {w|w ∈ A, {v|v ∈ A, v < w} = ∅}. If w^{1}, ⋯, w^{k} ∈ T and w^{i} ≠ w^{j} for any i ≠ j, then (M2) implies that μ(∪_{i=1}^{k}K_{w^{i}}) = ∑_{i=1}^{k} μ(K_{w^{i}}) ≥ kα. Hence k ≤ 1/α. So T is a finite set. Set M = max_{w∈T} |w| and choose w ∈ A with |w| > M. Then there exists a sequence {w(i)}_{i=1,2,...} ⊂ A such that w(1) = w and w(1) ≥ w(2) ≥ w(3) ≥ . . .. Let x = π(ω), where π : Σ → K is given in Definition 1.2.1 and ω ∈ Σ is the unique infinite sequence contained in ∩_{i=1,2,...}Σ_{w(i)}. Then μ({x}) = lim_{i→∞} μ(K_{w(i)}) ≥ α. This contradicts to (M3). Hence we have verified (G2). □

Definition 1.2.9. Let μ ∈ M(K) and denote μ̄ = μ/μ(K). Then gμ̄ defined in Proposition 1.2.8 is called the gauge function on W_{∗} induced by the measure μ. If no confusion can occur, we use μ to denote gμ̄ and write Λ_{s}(μ) = Λ_{s}(gμ̄). {Λ_{s}(μ)}_{0<s≤1} is called a scale on Σ induced by the measure μ.

The following two facts are immediate from the definition.

Proposition 1.2.10. Let μ be a self-similar measure. Then the scale induced by μ is an self-similar scale with the same weight as μ.

Proposition 1.2.11. Let μ ∈ M(K). Then the scale induced by μ is elliptic if and only if μ is elliptic.

1.3. Volume doubling property

In this section, S is a finite set, L = (K, S, {F_{i}}_{i∈S}) is a self-similar structure with K ≠ \overline{V}_{0}. Also 𝒮 = \{Λ_{s}\}_{0<s≤1} is a right-continuous scale on Σ and μ is always assumed to be a Borel regular finite measure on K which belongs to M(K). We will introduce a system of neighborhoods {U_{s}^{(n)}(x)} of x ∈ K associated with the scale Σ and consider the counterpart of “volume doubling measures” on K.

Ordinarily, if (X, d) is a metric space and μ is a Borel measure on (X, d), then μ is said to have the volume doubling property (or μ is volume doubling in short) if μ(B_{2r}(x)) ≤ Cμ(B_{r}(x)) for any x ∈ X and r > 0, where B_{r}(x) is a ball B_{r}(x) = {y|d(x, y) < r} and C is a constant which is independent of x and r. We will think of U_{s}^{(n)}(x) as a ball and introduce the notion corresponding to the volume doubling property. The main goal of this section, which is Theorem 1.3.5, is to establish conditions which are equivalent to the volume doubling property in our framework.

To start with, we associate subsets of words with those of self-similar sets.

Definition 1.3.1. Let Γ ⊆ W_{∗} and let A ⊆ K.

(1) W (Γ, A) = {w|w ∈ Γ, K_{w} ∩ A ≠ ∅}.

(2) K(Γ) = ∪_{w∈Γ}K_{w}.

(3) Define W^{(n)}(Γ, A) and K^{(n)}(Γ, A) by W^{(0)}(Γ, A) = W (Γ, A), K^{(n)}(Γ, A) = K(W^{(n)}(Γ, A)) and W^{(n+1)}(Γ, A) = W (Γ, K^{(n)}(Γ, A)) for n = 0, 1, . . ..

(4) We use ∂A be the topological boundary of A, i.e. ∂A = \overline{A^{c}} ∩ \overline{A}.

Under a scale 𝒮 = {Λ_{s}}_{0<s≤1}, the “radius” of K_{w} is thought of as s if w ∈ Λ_{s}. In this way we may define a ball of radius s with respect to a scale in the following way.

Definition 1.3.2. Let 𝒮 = {Λ_{s}}_{0<s≤1} be a scale on Σ. For x ∈ K, we write Λ_{s,x}^{n} = W^{(n)}(Λ_{s}, {x}) and U_{s}^{(n)}(x) = K^{(n)}(Λ_{s}, {x}) for n ≥ 0. In particular, we use Λ_{s,x} = Λ_{s,x}^{0}, K_{s}(x) = U_{s}^{(0)}(x) and U_{s}(x) = U_{s}^{(1)}(x). Also set Λ_{s,w} = W (Λ_{s}, K_{w}) for w ∈ W_{∗}.

U_{s}^{(n)}(x) is a neighborhood of x for any n. In the case n = 0, however, K_{s}(x) = U_{s}^{(0)}(x) is not a good ball with center x since x may be very close to the boundary of K_{s}(x) i.e. ∂K_{s}(x). (Note that if x ∈ K_{w}\F_{w}(V_{0}) and w ∈ Λ_{s}, then K_{s}(x) = K_{w}.) This will make a crucial difference between the role of {U_{s}^{(n)}(x)}_{x∈K,s>0} for n = 0 and that for n ≥ 1.

Definition 1.3.3. Let 𝒮 = {Λ_{s}}_{0<s≤1} be a scale on Σ and let μ ∈ M(K). For n ≥ 0, we define a property (VD)_{n} on (𝒮, μ) as follows.

(VD)_{n} There exist α ∈ (0, 1) and c_{V} > 0 such that μ(U_{s}^{(n)}(x)) ≤ c_{V} μ(U_{αs}^{(n)}(x)) for any s ∈ (0, 1] and any x ∈ K.

If (𝒮, μ) satisfy (VD)_{n} for some n ≥ 1, we say that μ has the volume doubling property (or (VD) for short) with respect to 𝒮.

If 𝒮 satisfies (EL1), then (VD)_{n} will be shown to be equivalent to (VD)_{1} for any n ≥ 1 in Theorem 1.3.11. Therefore, μ has (VD) with respect to 𝒮 if and only if (VD)_{n} holds for all n ≥ 1.

We introduce several notions to describe the conditions which is equivalent to the volume doubling property.

Definition 1.3.4. (1) Let φ : W_{∗} → [0, ∞). We say that φ is gentle with respect to (L, 𝒮) if and only if it satisfies the following condition (GE):

(GE) There exists c_{G} > 0 such that φ(w) ≤ c_{G}φ(v) for any s ∈ (0, 1] and any w, v ∈ Λ_{s} with K_{w} ∩ K_{v} ≠ ∅.

μ is said to be gentle with respect to 𝒮 if and only if φ_{μ} is gentle with respect to (L, 𝒮), where φ_{μ} is defined by φ(w) = μ(K_{w}).

(2) 𝒮 is said to be locally finite with respect to L if and only if it satisfies the following condition (LF):

(LF) sup{#(Λ_{s,x}^{1})|s ∈ (0, 1], x ∈ K} < +∞,

(3) Let n ∈ ℕ. We define properties (A)_{n} on (𝒮, μ) as follows.

(A)_{n} There exists c_{A} > 0 such that μ(U_{s}^{(n)}(x)) ≤ c_{A}μ(K_{s}(x)) for any s ∈ (0, 1] and any x ∈ K.

Note that the notion of “gentle measure” concerns both a scale and a measure while the condition (LF) is determined solely by a scale.

Theorem 1.3.5. Assume that 𝒮 = {Λ_{s}}_{0<s≤1} is elliptic. Let n ≥ 1. Then the following three conditions are equivalent.

(1) 𝒮 is locally finite and μ is elliptic and gentle with respect to 𝒮. In short, (LF) ∧ (ELm) ∧ (GE).

(2) (𝒮, μ) has properties (A)_{n} and (VD)_{0}. In short, (A)_{n} ∧ (VD)_{0}.

(3) (𝒮, μ) satisfies (VD)_{n}.

In particular, (VD)_{n} is equivalent to (VD)_{1} for any n ∈ ℕ and (VD) ⇔ (LF) ∧ (ELm) ∧ (GE).

Remark. We will show stronger statement on the equivalence between (1) and (2). In fact, by Theorems 1.3.8 and 1.3.10, (LF) ∧ (GE) ⇔ (A)_{n} and (ELm) ⇔ (VD)_{0}.

The main purpose of the rest of this section is to prove the above theorem. First we examine the condition (LF).

Lemma 1.3.6. The following three conditions are equivalent:

(1) 𝒮 is locally finite with respect to L.

(2) sup{#(Λ_{s,w})|s ∈ (0, 1], w ∈ Λ_{s}} < +∞.

(3) sup{#(Λ_{s,x}^{n})|s ∈ (0, 1], x ∈ K} < +∞ for any n ≥ 1.

Moreover, if 𝒮 is locally finite with respect to L, then L is strongly finite.

Proof. (1) ⇒ (2): Let s ∈ (0, 1] and let w ∈ Λ_{s}. Choose x ∈ K_{w}. Then Λ_{s,w} ⊆ Λ_{s,x}^{1}. This immediately implies (2).

(2) ⇒ (3): Set M = sup{#(Λ_{s,w})|s ∈ (0, 1], w ∈ Λ_{s}}. First we show that (2) implies that L is strongly finite. For any x ∈ K, if #(π^{−1}(x)) ≥ m, then #(Λ_{s,x}) ≥ m for sufficiently small s. Choose w ∈ Λ_{s,x}, then #(Λ_{s,w}) ≥ m. Therefore (2) implies that #(π^{−1}(x)) ≤ M and #(Λ_{s,x}) ≤ M. Note Λ_{s,x}^{n} = ∪_{w∈Λ_{s,x}^{n−1}}Λ_{s,w}. Hence #(Λ_{s,x}^{n}) ≤ M^{n+1} for any n ≥ 1.

(3) ⇒ (1): Obvious. □

If μ satisfy (VD)_{n}, then μ(U_{s}^{(n)}(x)) ≤ (c_{V})^{m}μ(U_{α^{m}s}^{(n)}(x)) for any m ≥ 1. This fact lead us to the following proposition.

Proposition 1.3.7. Let n ≥ 0. (VD)_{n} is equivalent to the following stronger condition: For any α ∈ (0, 1), there exists c > 0 such that μ(U_{s}^{(n)}(x)) ≤ cμ(U_{αs}^{(n)}(x)) for any s ∈ (0, 1] and any x ∈ K.

Now we give the first piece of a proof of Theorem 1.3.5.

Theorem 1.3.8. Let n ∈ ℕ. Then μ is gentle with respect to the scale 𝒮 and 𝒮 is locally finite if and only if the property (A)_{n} is satisfied. In short, (GE) ∧ (LF) ⇔ (A)_{n}. In particular, (A)_{n} and (A)_{m} are equivalent to each other for any n, m ∈ ℕ.

Proof. (GE) ∧ (LF) ⇒ (A)_{n}: For any w ∈ Λ_{s,x}^{n}, there exist w^{0}, w^{1}, …, w^{n} ∈ Λ_{s} such that w^{0} ∈ Λ_{s,x}, w^{n} = w and K_{w^{j−1}} ∩ K_{w^{j}} ≠ ∅ for j = 1, …, n. Hence by (GE),

μ(U_{s}^{(n)}(x)) = ∑_{w∈Λ_{s,x}^{n}} μ(K_{w}) ≤

(c_{G})^{n}#(Λ_{s,x}^{n}) max_{w∈Λ_{s,x}} μ(K_{w}) ≤ (c_{G})^{n}#(Λ_{s,x}^{n})μ(K_{s}(x)).

Therefore by Lemma 1.3.6, (LF) implies (A)_{n}. (A)_{n} ⇒ (GE): Note that (A)_{n} implies (A)_{1}. Let w, v ∈ Λ_{s} satisfy K_{w} ∩ K_{v} ≠ ∅. Since K\\overline{V}_{0} ≠ ∅, K_{w}\F_{w}(V_{0}) ≠ ∅. Choose x ∈ K_{w}\F_{w}(V_{0}). Then K_{s}(x) = K_{w}. By (A)_{1}, c_{A}μ(K_{w}) = c_{A}μ(K_{s}(x)) ≥ μ(U_{s}(x)) ≥ μ(K_{v}). (A)_{n} ∧ (GE) ⇒ (LF): Let w ∈ Λ_{s}. Choosing x ∈ K_{w}\F_{w}(V_{0}), we see that K_{s}(x) = K_{w} and Λ_{s,x}^{1} = Λ_{s,w}. By (A)_{1} and (GE),

c_{A}μ(K_{s}(x)) ≥ μ(U_{s}(x)) ≥ c_{G}#(Λ_{s,x}^{1})μ(K_{w}) = c_{G}#(Λ_{s,x}^{1})μ(K_{s}(x)).

Dividing this by μ(K_{s}(x)), we obtain (LF). □

The second piece of Theorem 1.3.5 is the equivalence between (VD)_{0} and (ELm). To give an exact statement, we need a definition.

Definition 1.3.9. (L, 𝒮, μ) is said to have the property (ELmg) if and only if the following condition is satisfied: There exist α ∈ (0, 1) and c_{E} > 0 such that μ(K_{wv}) ≥ c_{E}μ(K_{w}) for any s ∈ (0, 1], any w ∈ Λ_{s} and any v ∈ W_{∗} with wv ∈ Λ_{αs}.

Remark. If (ELmg) is satisfied, then, for any β ∈ (0, 1), there exists c > 0 such that μ(K_{wv}) ≥ cμ(K_{w}) for any s ∈ (0, 1), any w ∈ Λ_{s} and any v ∈ W_{∗} with wv ∈ Λ_{βs}. Indeed, for any k > 0, μ(K_{wv}) ≥ α^{k}μ(K_{w}) for any s ∈ Λ_{s}, any w ∈ Λ_{s} and any v ∈ W_{∗} with wv ∈ Λ_{α^{k}s}.

Theorem 1.3.10. (VD)_{0} ⇔ (ELmg) ⇔ (ELm) ∧ (EL2). In particular, if 𝒮 is elliptic, then (VD)_{0} is equivalent to (ELm).

Proof. (VD)_{0} ⇒ (ELmg): Let w ∈ Λ_{s} and let wv ∈ Λ_{αs}. Choose x ∈ K_{wv}\F_{wv}(V_{0}). Then K_{αs}(x) = K_{wv}. Hence by (VD)_{0}, c_{B}μ(K_{wv}) = c_{B}μ(K_{αs}(x)) ≥ μ(K_{s}(x)). Since K_{s}(x) ⊇ K_{w}, c_{B}μ(K_{wv}) ≥ μ(K_{w}). This shows (ELmg). (ELmg) ⇒ (ELm) ∧ (EL2): Let g be the gauge function of the scale 𝒮. For any w = w_{1} … w_{m} ∈ W_{∗}, there exists n ≥ 0 such that g(w) = g(w_{1} … w_{m−n}) < g(w_{1} … w_{m−n−1}). Set w′ = w_{1} … w_{m−n}. Let s = g(w). Note that w′ ∈ Λ_{s}. For any i ∈ S, we can find v ∈ W_{∗} which satisfies wiv ∈ Λ_{αs}. By (ELmg),

μ(K_{wi}) ≥ μ(K_{wiv}) ≥ cμ(K_{w′}) ≥ cμ(K_{w}).

This shows (ELm). Recall the remark after Definition 1.2.5. Under (ELm), there exist δ ∈ (0, 1) and k ≥ 1 such that μ(K_{wv}) ≤ δμ(K_{w}) for any w ∈ W_{∗} and any v ∈ W_{k}. Therefore (ELmg) implies

c_{E}μ(K_{w}) ≤ μ(K_{wv}) ≤ δ^{[|v|/k]}μ(K_{w}).

for any s ∈ (0, 1], any w ∈ Λ_{s} and any v ∈ W_{∗} with wv ∈ Λ_{αs}, where [x] is the integral part of x. Dividing this by μ(K_{w}), we have uniform upper estimate of |v|. This shows (EL2). (ELm) ∧ (EL2) ⇒ (VD)_{0}: Let x ∈ K. For any w ∈ Λ_{s,x}, we may choose v(w) ∈ W_{∗} so that wv(w) ∈ Λ_{αs}. By (EL2), we have |v(w)| ≤ m, where m is independent of x, s and w. Using (ELm), we obtain μ(K_{wv(w)}) ≥ c^{m}μ(K_{w}). Hence,

μ(K_{αs}(x)) ≥ ∑_{w∈Λ_{s,x}} μ(K_{wv(w)}) ≥ ∑_{w∈Λ_{s,x}} c^{m}μ(K_{w}) = c^{m}μ(K_{s}(x)).

Therefore we have (VD)_{0}. □

The next theorem is a generalized version of Theorem 1.3.5.

Theorem 1.3.11. Let n ≥ 1. Assume that 𝒮 satisfies (EL1). Then the following three conditions are equivalent:

(1) 𝒮 is locally finite, μ is gentle with respect to 𝒮 and satisfy (VD)_{0}. In short, (LF) ∧ (GE) ∧ (VD)_{0}.

(2) (𝒮, μ) has properties (A)_{n} and (VD)_{0}.

(3) (𝒮, μ) satisfies (VD)_{n}.

In particular, (VD)_{n} is equivalent to (VD)_{1} for any n ∈ ℕ and (VD) ⇔ (LF) ∧ (GE) ∧ (VD)_{0}.

To prove the above theorem, we need the following lemma.

Lemma 1.3.12. Let n ∈ ℕ. If 𝒮 satisfy (EL1), then there exist α ∈ (0, 1) and z ∈ K such that K_{w} ⊇ U_{αs}^{(n)}(F_{w}(z)) for any s ∈ (0, 1] and any w ∈ Λ_{s}.

Proof. Choose z ∈ K\\overline{V}_{0}. Since K\\overline{V}_{0} is open, there exists β ∈ (0, 1) such that U_{β}^{(n)}(z) ⊆ K\\overline{V}_{0}. Set m = max_{v∈Λ_{β}} |v|. Let w ∈ Λ_{s}. Note that (EL1) implies that |v| ≥ m for any wv ∈ Λ_{(α_{1})^{m}s}, where α_{1} is the constant appeared in Definition 1.1.11. Denote x = F_{w}(z) and α = (α_{1})^{m}. For any τ ∈ Λ_{αs,x}^{n}, there exist w^{0}, w^{1}, …, w^{k} ∈ Λ_{αs,x}^{n} such that k ≤ n, w^{k} = τ, w^{j} ∈ Λ_{αs,x}^{j} for j = 0, 1, …, k, where Λ_{αs,x}^{0} = Λ_{αs,x}, and K_{w^{j−1}} ∩ K_{w^{j}} ≠ ∅ for any j = 1, …, k. Let p = max{j|w^{j} ≤ w}. Then, w^{j} = wv^{j} for j = 0, 1, …, p. Since |v^{j}| ≥ m, there exists u^{j} ∈ Λ_{β} such that v^{j} ≤ u^{j}. It follows that z ∈ K_{v^{0}} ⊆ K_{u^{0}} and that K_{u^{j−1}} ∩ K_{u^{j}} ≠ ∅ for any j = 1, …, p. Therefore, u^{j} ∈ Λ_{β,z}^{j} for j = 1, …, p. This implies that K_{w^{j}} ⊆ F_{w}(U_{β}^{j}(z)) for j = 1, …, p. Hence K_{w^{p}} ∩ F_{w}(\overline{V}_{0}) = ∅. If p < k, then there exists w′ ∈ Λ_{s} such that w′ ≠ w and w^{p+1} ≤ w′. Since K_{w} ∩ K_{w′} = F_{w}(V_{0}) ∩ F_{w′}(V_{0}), we have K_{w^{p}} ∩ F_{w}(V_{0}) ≠ ∅. This is a contradiction and hence we have p = k. Hence K_{τ} ⊆ K_{w}. Therefore, U_{αs}^{(n)}(x) ⊆ K_{w}. □

Proof of Theorem 1.3.11. (LF) ∧ (GE) ∧ (VD)_{0} ⇒ (A)_{n} ∧ (VD)_{0}: This is obvious by Theorem 1.3.8. (A)_{n} ∧ (VD)_{0} ⇒ (VD)_{n}: For s ∈ (0, 1] and x ∈ K,

μ(U_{s}^{(n)}(x)) ≤ c_{A}μ(K_{s}(x)) ≤ c_{A}c_{B}μ(K_{αs}(x)) ≤ c_{A}c_{B}μ(U_{αs}^{(n)}(x)).

(VD)_{n} ⇒ (GE): By Lemma 1.3.12, there exist α ∈ (0, 1) and z ∈ K such that K_{w} ⊇ U_{αs}^{(n)}(F_{w}(z)) for any s ∈ (0, 1] and any w ∈ Λ_{s}. Proposition 1.3.7 implies that μ(U_{s}^{(n)}(x)) ≤ cμ(U_{αs}^{(n)}(x)). Now assume that w ≠ v ∈ Λ_{s} and K_{w} ∩ K_{v} ≠ ∅. Set x = F_{w}(z). Then

(1.3.1)  μ(K_{v}) ≤ μ(U_{s}^{(n)}(x)) ≤ cμ(U_{αs}^{(n)}(x)) ≤ cμ(K_{w}).

Hence, μ is gentle. (VD)_{n} ⇒ (VD)_{0}: Fix β ∈ (0, 1). Let w ∈ Λ_{s}. By Lemma 1.3.12, K_{wv} ⊃ U_{αβs}^{(n)}(x)

for any wv ∈ Λ_{βs}, where x = F_{wv}(z). Note that K_{w} ⊂ U_{s}^{(n)}(x). By Proposition 1.3.7, there exists c > 0 such that μ(U_{s}^{(n)}(x)) ≤ cμ(U_{αβs}^{(n)}(x)). Hence cμ(K_{wv}) ≥

cμ(U_{αβs}^{(n)}(x) ≥ μ(U_{s}^{(n)}(x)) ≥ μ(K_{w}). By Theorem 1.3.10, we have (VD)_{0}. (VD)_{n} ⇒ (LF): Let s ∈ (0, 1] and let w ∈ Λ_{s}. Choose x = F_{w}(z), where z is given in Lemma 1.3.12. Then by (1.3.1), using the similar argument as in the proof of Theorem 1.3.8, we see that μ(K_{v}) ≥ c^{−n}μ(K_{w}) for any v ∈ Λ_{s,x}^{n}. Hence

cμ(K_{w}) ≥ cμ(U_{αs}^{(n)}(x)) ≥ μ(U_{s}^{(n)}(x)) = ∑_{v∈Λ_{s,x}^{n}} μ(K_{v}) ≥ c^{−n}#(Λ_{s,x}^{n})μ(K_{w}).

Dividing this by μ(K_{w}), we obtain (LF) by Lemma 1.3.6. □

Finally combining Theorems 1.3.10 and 1.3.11, we immediately obtain Theorem 1.3.5.

1.4. Locally finiteness and gentleness

In this section, we will define the notion of a scale being gentle with respect to another scale. It will turn out that the relation of “being gentle with respect to” is an equivalence relation among elliptic scales and the locally finiteness is inherited from a scale to another scale by this equivalence relation. As in the previous section, we fix a self-similar structure L = (K, S, \{F_{i}\}_{i∈S}) with K ≠ \overline{V_{0}}. Also all the scales are assumed to be right continuous.

Definition 1.4.1. Let 𝒮_{1} and 𝒮_{2} be scales on Σ. 𝒮_{2} is said to be gentle with respect to 𝒮_{1} if and only if the gauge function of 𝒮_{2} is gentle with respect to (L, 𝒮_{1}).

Remark. Note that we need information on the self-similar structure L to determine whether 𝒮_{2} is gentle with respect to 𝒮_{1} or not.

Naturally we have the next proposition.

Proposition 1.4.2. Let 𝒮 be a scale and let μ ∈ M(K). Then μ is gentle with respect to 𝒮 if and only if {Λ_{s}(μ)}_{0<s≤1} is gentle with respect to 𝒮.

Here is the main results of this section.

Theorem 1.4.3. Let 𝒮_{1} and 𝒮_{2} be elliptic scales on Σ. Assume that 𝒮_{2} is gentle with respect to 𝒮_{1}.

(1) 𝒮_{1} is gentle with respect to 𝒮_{1}.

(2) If 𝒮_{1} is locally finite, then 𝒮_{2} is locally finite.

(3) 𝒮_{1} is gentle with respect to 𝒮_{2}.

(4) Let 𝒮_{3} be an elliptic scale on Σ. Suppose 𝒮_{3} is gentle with respect to 𝒮_{2}, then 𝒮_{3} is gentle with respect to 𝒮_{1}.

Proof. Let 𝒮_{1} = {Λ_{s}}_{0<s≤1} and let 𝒮_{2} = {Γ_{s}}_{0<s≤1}. Let l and g be the gauge functions of 𝒮_{1} and 𝒮_{2} respectively. First we show (1). Recall that w = w_{1} … w_{m} ∈ Λ_{s} if and only if l(w) ≤ s < l(w_{1} … w_{m−1}). By Proposition 1.1.12-(1), there exists c > 0 such that l(w) ≥ cl(w_{1} … w_{m−1}) for any w ∈ W_{∗}. Hence if w ∈ Λ_{s}, then l(w) ≤ s < l(w)/c. This shows that cl(w) ≤ l(v) for any w, v ∈ Λ_{s}. Hence we obtain (1).

Proofs of (2), (3) and (4) are based on the same idea. If w = w_{1} … w_{m} ∈ Γ_{s}, then

g(w_{1} … w_{m−1}) > s ≥ g(w).

On the other hand, there exists a unique k ≤ m such that l(w_{1} … w_{k−1}) > l(w_{1} … w_{k}) = l(w_{1} … w_{m}). By (EL2), m − k ≤ n, where n ∈ ℕ is independent of w. Let a = l(w) and let w′ = w_{1} … w_{k}. Since 𝒮_{2} is gentle with respect to 𝒮_{1}, g(v) ≤ cg(w′) for any v ∈ Λ_{a,w′}. If 𝒮_{1} is elliptic, g(w′) ≤ (β_{1})^{−n}g(w), where β_{1} is the constant appearing in Proposition 1.1.12-(1). Therefore there exists c′ > 0 such that g(v) ≤ c′g(w) for any v ∈ Λ_{a,w′}. Using Proposition 1.1.12-(2), we see that there exists p ∈ ℕ such that g(vτ) ≤ g(w) ≤ s for any τ ∈ W_{p}. (Note that p in independent of s and w.) This shows that, for any τ ∈ W_{p} and any v ∈ Λ_{a,w′}, there exists a unique v′ such that vτ ≤ v′ and v′ ∈ Γ_{s}. Define π(vτ) = v′. Then π : Λ_{a,w′} × W_{p} → Γ_{s}. Note that Γ_{s,w} is included in the image of π. Hence #(Γ_{s,w}) ≤ #(Λ_{a,w′})N^{p}, where N = #(S). By Lemma 1.3.6, if 𝒮_{1} is locally finite, then so is 𝒮_{2}. This proves (2). Next we show (3). For any v′ ∈ Γ_{s,w}, choose v ∈ Λ_{a,w′} and τ ∈ W_{p} so that π(vτ) = v′. Then l(vτ) ≤ l(v′). Since 𝒮_{1} satisfies (EL1), there exists γ > 0 such that l(wi) ≥ γl(w) for any w ∈ W_{∗} and any i ∈ S. Therefore,

l(v′) ≥ l(vτ) ≥ γ^{p}l(v) ≥ γ^{p}a = γ^{p}l(w).

Hence 𝒮_{1} is gentle with respect to 𝒮_{2}.

To prove (4), we write 𝒮_{3} = {Ω_{s}}_{0<s≤1}. Let w ∈ Ω_{s}. There exist k, j ∈ ℕ such that j ≤ k ≤ m = |w|, g(w_{1} … w_{k−1}) > g(w_{1} … w_{k}) = g(w) and l(w_{1} … w_{j−1}) > l(w_{1} … w_{j}) = l(w_{1} … w_{k}). Now using the same construction as π above, we have maps π_{1} : Λ_{a,w′′} × W_{p} → Γ_{b} and π_{2} : Γ_{b,w′} × W_{q} → Ω_{s}, where w′′ = w_{1} … w_{j}, a = l(w′′), w′ = w_{1} … w_{k} and b = g(w′), satisfying the same properties as π. Now for any v ∈ Ω_{s,w}, there exist v′ ∈ Γ_{b,w′}, τ′ ∈ W_{q}, v′′ ∈ Λ_{a,w′′} and τ′′ ∈ W_{p} such that π_{2}(v′τ′) = v and π_{1}(v′′τ′′) = v′. Note that v′τ′ ≤ v and v′′τ′′ ≤ v′. This implies

l(v) ≥ l(v′τ′) ≥ γ^{q}l(v′) ≥ γ^{q}l(v′′τ′′) ≥ γ^{p+q}l(v′′) ≥ γ^{p+q}a ≥ γ^{p+q}l(w)

This shows that 𝒮_{1} is gentle with respect to 𝒮_{3}. Applying (3), we obtain the desired result. □

By the above theorem, the relation “gentle with respect to” is an equivalence relation on elliptic scales.

Definition 1.4.4. (1) Let 𝒮_{1} and 𝒮_{2} be elliptic scales. We write 𝒮_{1} ∼_{GE} 𝒮_{2} if and only if 𝒮_{1} is gentle with respect to 𝒮_{2}.

(2) Let 𝒮 be a scale. We define

M_{VD}(L, 𝒮) = {μ|μ ∈ M(K), μ has (VD) with respect to 𝒮}.

Proposition 1.4.5. (1) Let 𝒮 be an elliptic scale on Σ. If M_{VD}(L, 𝒮) ≠ ∅, then 𝒮 is locally finite.

(2) Let 𝒮_{1} and 𝒮_{2} be elliptic scales. Then

M_{VD}(L, 𝒮_{1}) ∩ M_{VD}(L, 𝒮_{2}) ≠ ∅ ⇒ 𝒮_{1} ∼_{GE} 𝒮_{2} ⇒ M_{VD}(L, S_{1}) = M_{VD}(L, S_{2}).

Proof. (1) This is immediate by Theorem 1.3.5.

(2) Let μ ∈ M_{VD}(L, 𝒮_{1}) ∩ M_{VD}(L, 𝒮_{2}). If 𝒮_{3} is the scale induced by μ, then 𝒮_{1} ∼_{GE} 𝒮_{3} and 𝒮_{2} ∼_{GE} 𝒮_{3} by Proposition 1.4.2. Hence 𝒮_{1} ∼_{GE} 𝒮_{2}. Next assume

𝒮_{1} ∼_{GE} 𝒮_{2} and let μ ∈ M_{VD}(L, 𝒮_{1}). Let 𝒮_{3} be the scale induced by μ. Then 𝒮_{1} ∼_{GE} 𝒮_{3} by Proposition 1.4.2. Hence 𝒮_{2} ∼_{GE} 𝒮_{3}. Again by Proposition 1.4.2,

μ ∈ M_{VD}(L, 𝒮_{2}). Hence M_{VD}(L, 𝒮_{1}) ⊆ M_{VD}(L, 𝒮_{2}). Exchanging 𝒮_{1} and 𝒮_{2}, we see M_{VD}(L, 𝒮_{1}) = M_{VD}(L, 𝒮_{2}). □

Denote the collection of elliptic scales on Σ by E𝒮(Σ). Then, by the above results, an equivalence class of E𝒮(Σ)/∼_{GE} tells us whether a scale 𝒮 is locally finite

or not and determines M_{VD}(L, 𝒮), the family of volume doubling measures with respect to 𝒮. Those facts raises our curiosity on the structure of E𝒮(Σ)/∼_{GE}. In the following sections, we will study this problem in a restricted situation.

We conclude this section by giving an important necessary condition for two self-similar scales being gentle.

Notation. For w ∈ W_{#} and any n ∈ ℕ, we define (w)^{n} = \underbrace{w … w}_{n times} ∈ W_{∗}. Also

(w)^{∞} = www … ∈ Σ.

Lemma 1.4.6. Let a = (a_{i})_{i∈S} ∈ (0, 1)^{S} and let b = (b_{i})_{i∈S} ∈ (0, 1)^{S}. Assume

that 𝒮(a) ∼_{GE} 𝒮(b). If w, w′, v, v′ ∈ W_{#} and π(v(w)^{∞}) = π(v′(w′)^{∞}), then \begin{matrix}log & a_{w} \\ log & b_{w} \end{matrix} =

\begin{matrix}log & a_{w′} \\ log & b_{w′} \end{matrix}.

Proof. For sufficiently small s, there exist i(s), j(s) ∈ ℕ and w(s), w′(s) ∈ W_{∗} such that w < w(s), w′ < w′(s), v(w)^{i(s)}w(s), v′(w′)^{j(s)}w′(s) ∈ Λ_{s}(a) Set v(s) = v(w)^{i(s)}w(s) and v′(s) = v′(w′)^{j(s)}w′(s). By Lemma 1.1.13, a_{v(s)}/a_{v′(s)} is uniformly bounded with respect to s. Since a_{v}a_{w(s)}/(a_{v′}a_{w′(s)}) is uniformly bounded, we see that (a_{w})^{i(s)}/(a_{w′})^{j(s)} is uniformly bounded with respect to s. As 𝒮(a) ∼_{GE} 𝒮(b),

we see that b_{v(s)}/b_{v′(s)} is uniformly bounded as well. Note that b_{v}b_{w(s)}/(b_{v′}b_{w′(s)}) is uniformly bounded. Hence

\frac{(b_{w})^{i(s)}}{(b_{w′})^{j(s)}} = (a_{w})^{i(s)(α−β)} \frac{(a_{w})^{i(s)β}}{(a_{w′})^{j(s)β}} ,

where α = log b_{w}/ log a_{w} and β = log b_{w′}/ log a_{w′}, is uniformly bounded. Since i(s) → +∞ as s ↓ 0, it follows that α = β. □

1.5. Rationally ramified self-similar sets 1

In this section, we will introduce a special class of self-similar structures called “rationally ramified self-similar structures”. L = (K, S, {F_{i}}_{i∈S}) is assumed to be a self-similar structure throughout this section. Roughly speaking, L is called rationally ramified if K_{i}∩K_{j} is again a self-similar set. This class of self-similar sets include post critically finite self-similar sets, for example, the Sierpinski gasket, as well as so called “infinitely ramified” self-similar sets like the Sierpinski carpet and the Menger sponge. The advantage of a rationally ramified self-similar structure is that one can give simple characterizations for the locally finiteness of a scale and the gentleness of two scales. Using such results, we can explicitly determine the class of self-similar measures which have the volume doubling property with respect to a given scale for rationally ramified self-similar sets. See the next section for details

To start with, we need several notions and results on the shift space.

Definition 1.5.1. Let X be a non-empty finite subset of W_{#}. For w ∈ W_{#}, we denote w = (w)_{1} … (w)_{|w|}, where (w)_{i} ∈ S for i = 1, …, |w|. We define a map ι_{X} from Σ(X) = {x_{1}x_{2} … |x_{i} ∈ X for any i ∈ ℕ} to Σ(S) by

ι_{X}(x_{1}x_{2} …) = (x_{1})_{1} … (x_{1})_{|x_{1}|}(x_{2})_{1} … (x_{2})_{|x_{2}|} ….

Define Σ[X] = ι_{X}(Σ(X)), K[X] = π(Σ[X]), Σ_{w}[X] = σ_{w}(ι_{X}(Σ(X))) and K_{w}[X] = F_{w}(K[X])(= π(Σ_{w}[X])) for w ∈ W_{#}. X is called independent if and only if ι_{X} is injective. When X is independent, we sometimes identify Σ(X) with Σ[X].

For example, let S = {1, 2} and let X = {1, 12, 21} Then X is not independent. In fact,

2112(1)^{∞} = ι_{X}(cb(a)^{∞}) = ι_{X}(cac(a)^{∞}).

where a = 1, b = 12, c = 21.

Since ι_{X} : Σ(X) → Σ[X] and π : Σ → K are continuous, we have the following lemma.

Lemma 1.5.2. If X is a nonempty finite subset of W_{∗}, then Σ[X] and K[X] are compact.

Now we study how to characterize the independence of X.

Definition 1.5.3. Let X be a nonempty subset of W_{#}.

(1) For m ≥ 0, define ρ_{m} : Σ → W_{m} by ρ_{m}(ω) = ω_{1} … ω_{m} for ω = ω_{1}ω_{2} . . ..

(2) For m ≥ n, define ρ_{m,n} : W_{m} → W_{n} by ρ_{m,n}(w_{1} … w_{m}) = w_{1} … w_{n}.

(3) For x_{1}, x_{2}, …, x_{m} ∈ X, recalling that each x_{i} ∈ W_{∗}, we may regard x_{1} … x_{m} ∈ W_{∗}(X) as an element of W_{∗}. We use ι_{X}^{w}(x_{1} … x_{m}) to denote x_{1} … x_{m} as an element of W_{∗} to avoid confusion. In other word, ι_{X}^{w} : W_{∗}(X) → W_{∗} is defined by

ι_{X}^{w}(x_{1} … x_{m}) = x_{1}(1) … x_{1}(n_{1}) … x_{m}(1) … x_{m}(n_{m}),

where n_{i} = |x_{i}| and x_{i} = x_{i}(1)x_{i}(2) … x_{i}(n_{i}) ∈ W_{n_{i}} for i = 1, …, m.

(4) For m ≥ 0, we define

Q_{m}(X) = ⋃_{w,v∈X,w≠v} (ρ_{m}(Σ_{w}[X])) ∩ ρ_{m}(Σ_{v}[X])).

The following fact is immediate from the above definition. It says that an element of Q_{m}(X) can be expressed by two different words of X whose first symbols are different. X is assumed to be a nonempty finite subset of W_{#}.

Lemma 1.5.4. Let w ∈ W_{m}. Then w ∈ Q_{m}(X) if and only if there exist x_{1} … x_{k} and x′_{1} … x′_{n} ∈ W_{∗}(X) such that x_{1} ≠ x′_{1}, ι_{X}^{w}(x_{1} … x_{k}) ≤ w < ι_{X}^{w}(x_{1} … x_{k−1}) and ι_{X}^{w}(x′_{1} … x′_{n}) ≤ w < ι_{X}^{w}(x′_{1} … x′_{n−1}). Moreover, if m ≥ n, then ρ_{m,n}(Q_{m}(X)) ⊆ Q_{n}(X).

Lemma 1.5.5. Let ω ∈ Σ. Suppose that there exist w ∈ X and m_{1} < m_{2} < … such that ρ_{m_{i}}(ω) ∈ ρ_{m_{i}}(Σ_{w}[X]). Then ω = ι_{X}(wx_{2} …) for some x_{2}, x_{3}, … ∈ X.

Proof. For sufficiently large i, we may find x(i) ∈ X such that ρ_{m_{i}}(ω) ∈ Σ_{wx(i)}. Since X is a finite set, we may find x_{2} ∈ X and a subsequence {m_{2,i}}_{i≥1} of {m_{i}}_{i≥1} such that ρ_{m_{2,i}}(ω) ∈ Σ_{wx_{2}} for any i. Repeating the same procedure, we may inductively obtain x_{j} ∈ X and {m_{j,i}}_{i≥1} for j ≥ 2. Now, ω = ι_{X}(wx_{2}x_{3} …).

□

We have a simple characterization of the independence of X in terms of Q_{m}(X).

Theorem 1.5.6. Let X be a nonempty finite subset of W_{#}. Then X is independent if and only if Q_{m}(X) = ∅ for some m ∈ ℕ.

Remark. By Lemma 1.5.4, if Q_{m}(X) = ∅, then Q_{n}(X) = ∅ for any n ≥ m.

Proof. If X is not independent, then there exist x_{1}x_{2} …, x′_{1}x′_{2} … ∈ Σ(X) such that ι_{X}(x_{1}x_{2} …) = ι_{X}(x′_{1}x′_{2} …). We may assume that x_{1} ≠ x′_{1} without loss of generality. Now, ρ_{m}(ι_{X}(x_{1}x_{2} …)) ∈ Q_{m}(X) for any m ≥ 0.

Conversely suppose that Q_{m}(X) ≠ ∅ for any m ≥ 0. Set

Q_{m,n}^{∗}(X) = ρ_{m+n,m}(Q_{m+n}(X)).

Then {Q_{m,n}^{∗}(X)}_{n≥0} is a decreasing sequence of nonempty finite sets. Therefore, Q_{m}^{∗}(X) = ∩_{n≥0}Q_{m,n}^{∗}(X) is not empty. Also it follows that ρ_{k,l}(Q_{k}^{∗}(X)) = Q_{l}^{∗}(X) for any k, l ∈ ℕ with k ≥ l. Therefore, there exists ω = ω_{1}ω_{2} … ∈ Σ such that ρ_{m}(ω) ∈ Q_{m}^{∗}(X) for any m ≥ 0. For each m, there exist w(m), v(m) ∈ X such that w(m) ≠ v(m) and ρ_{m}(ω) ∈ ρ_{m}(ι_{X}(Σ(X))) ∩ Σ_{w(m)} ∩ Σ_{v(m)}. Since X is a finite set, there exist w, v ∈ X and {m_{i}}_{i≥1} such that w ≠ v, w(m_{i}) = w and v(m_{i}) = v. Now using Lemma 1.5.5, we see that ω = ι_{X}(wx_{2}x_{3} …) = ι_{X}(vx′_{2}x′_{3} …) for some {x_{i}}, {x′_{i}} ∈ X. Hence ι_{X} is not injective and therefore X is not independent. □

Hereafter, if a nonempty finite subset X of W_{#} is independent, we think of x_{1} … x_{m} ∈ W_{∗}(X) (where x_{i} ∈ X for any i) as an element of W_{∗} in the natural manner.

Before getting to the definition of rationally ramified self-similar structure, we still need several notions.

Definition 1.5.7. Let Σ_{0} be a nonempty subset of Σ and let x ∈ W_{∗}. We define O_{Σ_{0},x}(ω)

O_{Σ_{0},x}(ω) = #({m|m ≥ 0, σ^{m}ω ∈ σ_{x}(Σ_{0})})

for any ω ∈ Σ. We allow ∞ as a value of O_{Σ_{0},x}(ω).

The following two lemmas are basic facts on O_{Σ[X],x}(ω).

Lemma 1.5.8. Let X be a nonempty finite subset of W_{∗} and let x ∈ W_{∗}. If

sup_{ω∈Σ} O_{Σ[X],x}(ω) = +∞,

then there exists τ ∈ Σ such that O_{Σ[X],x}(τ) = ∞.

Proof. Set k = max_{w∈X} |w| and define M = |x|(k^{2} + 3). By the above assumption, we may choose ω ∈ Σ_{x}[X] so that #({m|σ^{m}(ω) ∈ Σ_{x}[X]}) ≥ M. Let ω = xι_{X}(x_{1}x_{2} …), where x_{1}, x_{2}, … ∈ X. There exists a sequence {m_{i}}_{0=1,...,k^{2}+2} such that m_{0} = 0, m_{i} + |x| ≤ m_{i+1} for any i = 1, …, k^{2} + 1 and σ^{m_{i}}(ω) ∈ Σ_{x}[X] for any i = 1, …, k^{2} + 2. Choose n so that m_{k^{2}+2} + |x| < |xx_{1} … x_{n−1}|. Then |x_{n+1} … x_{n+k}| ≥ k and so, for any i = 1, …, k^{2} + 2, there exists {x_{j}^{i}}_{j=1,...,n(i)} ⊂ X such that xx_{1} … x_{n}x_{n+1} … x_{n+k} < ω_{1} … ω_{m_{i}}xx_{1}^{i} … x_{n(i)}^{i} ≤ xx_{1} … x_{n}. Since k^{2} + 2 > |x_{n+1} … x_{n+k}| + 1, we see ω_{1} … ω_{m_{p}}xx_{1}^{p} … x_{n(p)}^{p} = ω_{1} … ω_{m_{q}}xx_{1}^{q} … x_{n(q)}^{q} for some 0 ≤ p < q ≤ k. (We set x_{i}^{0} = x_{i} and n(0) = n.) Note that m_{p} + |x| ≤ m_{q}. Hence we have l which satisfies σ^{l}(x_{1}^{p} … x_{n(p)}^{p}) = xx_{1}^{q} … x_{n(q)}^{q}. Set w = x_{1}^{p} … x_{n(p)}^{p} and define τ = (w)^{∞}. Then τ = (w)^{i}w_{1} … w_{l}xx_{1}^{q} … x_{n(q)}^{q}(w)^{∞} for any i. Therefore, σ^{|w|i+l}τ ∈ Σ_{x}[X] for any i. □

Remark. In the proof, we have shown the following statement: Let k = max_{w∈X} |w|. If O_{Σ[X],x}(ω) ≥ M = |x|(k^{2} + 3) for some ω ∈ Σ, then there exists τ ∈ Σ such that O_{Σ[X],x}(τ) = +∞.

As a final step to the definition of rationally ramified self-similar structures, we need several definitions.

Definition 1.5.9. Let Ω = (X, Y, φ, x, y), where X and Y are a non-empty independent finite subsets of W_{#}, φ is a bijective map between X and Y and x, y ∈ W_{#}.

(1) We define φ_{∗} : Σ_{x}[X] → Σ_{y}[Y] by φ_{∗}(xx_{1}x_{2} …) = yφ(x_{1})φ(x_{2}) … for any x_{1}, x_{2}, … ∈ X.

(2) A pair (ω, τ) ∈ Σ(S) × Σ(S) with ω ≠ τ is called a corresponding pair with respect to Ω if and only if ω = vω′ and τ = vφ_{∗}(ω′) for some v ∈ W_{∗} and some ω′ ∈ Σ_{x}[X].

(3) Ω is called a relation of L if and only if the first symbol of x is different from that of y, O_{Σ[X],x}(ω) and O_{Σ[Y],y}(ω) are finite for any ω ∈ Σ and π(ω) = π(τ) for any corresponding pair (ω, τ) with respect to Ω. The collection of relations of L is denoted by R_{L}.

(4) Let Ω = (X, Y, φ, x, y) be a relation of L. Ω′ = (X′, Y′, φ′, x, y) is called a sub-relation of Ω if X′ ⊆ X, Y′ = φ(X′) and φ′ = φ|_{X}.

(5) Let R ⊂ R_{L}. A relation Ω = (X, Y, φ, x, y) is said to be generated by R if there exists a sequence of sub-relations of relations in R, {(X_{i}, X_{i+1}, φ_{i}, x_{i}, x_{i+1})}_{i=1}^{m−1}, such that X = X_{1}, Y = X_{m}, x = x_{1}, y = x_{m} and φ = φ_{m−1}∘ … ∘φ_{1}. We use [R] to denote the collection of relations generated by R. If R ⊆ R′ ⊆ [R], then R′ is said to be generated by R or R is a generator of R′.

Remark. If (X, Y, φ, x, y) is a relation of L. Then sup_{w∈Σ} O_{Σ[X],x}(w) and sup_{w∈Σ} O_{Σ[Y],y}(w) is finite by Lemma 1.5.8.

Remark. If Ω = (X, Y, φ, x, y) be a relation of L, then so is (Y, X, φ^{−1}, y, x). We denote Ω^{−1} = (Y, X, φ^{−1}, y, x) and identify Ω with Ω^{−1}. In particular, if R is a subset of R_{L} for a self-similar structure L, then we always suppose that Ω^{−1} ∈ R for any Ω ∈ R. In making a list of elements of a relation set, we customary mention only one of (X, Y, φ, x, y) or (Y, X, φ^{−1}, y, x).

Definition 1.5.10 (Rationally ramified self-similar structure). A self-similar structure L = (K, S, {F_{i}}_{i∈S}) is said to be rationally ramified if and only if it is strongly finite and there exists a finite subset R of R_{L} satisfying the following property: for any i, j ∈ S with i ≠ j,

(1.5.1)  π^{−1}(K_{i} ∩ K_{j}) ∩ Σ_{i} = ⋃_{(X,Y,φ,x,y)∈R_{ij},x∈σ_{i}(W_{∗})} Σ_{x}[X],

where

R_{ij} = {Ω|Ω = (X, Y, φ, x, y) ∈ [R], x ∈ σ_{i}(W_{∗}), y ∈ σ_{j}(W_{∗})}.

R is called the relation set of L.

Note that [R] is a finite set if R is finite. We may assume that R = [R] in the above definition without loss of generality. However, as one will see in Example 1.5.12, R can be more simple than [R] in some cases.

Example 1.5.11 (the Sierpinski gasket). Let p_{1}, p_{2} and p_{3} be vertices of a regular triangle in ℂ. Define F_{i}(z) = (z − p_{i})/2 + p_{i} for i = 1, 2, 3. The Sierpinski gasket is the self-similar set with respect to {F_{1}, F_{2}, F_{3}}, i.e. K is the unique non-empty compact set satisfying K = F_{1}(K) ∪ F_{2}(K) ∪ F_{3}(K). L = (K, S, {F_{i}}_{i∈S}), where S = {1, 2, 3}, is a rationally ramified self-similar structure. Indeed, {({i}, {j}, φ_{ij}, j, i)|(i, j) = (1, 2), (2, 3), (3, 1)}, where φ_{ij}(i) = j, is a relation set. According to the convention in the remark above, this relation set contains 6 elements in fact.

Example 1.5.12 (the Sierpinski carpet). et p_{1} = 0, p_{2} = 1/2, p_{3} = 1, p_{4} = 1 + \sqrt{−1}/2, p_{5} = 1 + \sqrt{−1}, p_{6} = 1/2 + \sqrt{−1}, p_{7} = \sqrt{−1} and p_{8} = \sqrt{−1}/2. Define F_{i} : ℂ → ℂ by F_{i}(z) = (z − p_{i})/3 + p_{i} for i = 1, …, 8. Then there exists a unique nonempty compact subset to ℂ, K, which satisfies K = ∪_{i=1}^{8}F_{i}(K). K is called the Sierpinski carpet. Let L = (K, S, {F_{i}}_{i∈S}), where S = {1, …, 8}. Then L is a rationally ramified self-similar structure. To describe its relation set R, we let X_{1} = {1, 2, 3}, Y_{1} = {7, 6, 5}, φ_{1}(1) = 7, φ_{1}(2) = 6, φ_{1}(3) = 5, X_{2} = {1, 8, 7}, Y_{2} = {3, 4, 5}, φ_{2}(1) = 3, φ_{2}(8) = 4 and φ_{2}(7) = 5. Then

R = {(X_{1}, Y_{1}, φ_{1}, i, j)|(i, j) = (8, 1), (4, 3), (7, 8), (5, 4)}∪

{(X_{2}, Y_{2}, φ_{2}, i, j)|(i, j) = (2, 1), (6, 7), (3, 2), (5, 6)}, Figure 1.1. Sierpinski gasket

Figure 1.2. Sierpinski carpet

where φ_{3}(1) = 5 and φ_{4}(3) = 7. In this case, the set of relations generated by R, [R], is not equal to R. In fact,

[R] = R ∪ {({1}, {5}, φ_{15}, i, j), ({3}, {7}, φ_{37}, k, l)|

(i, j) = (6, 8), (4, 2), (k, l) = (8, 2), (6, 4)},

where φ_{mn} maps m to n. Those additions are really needed in the definition of rationally ramified self-similar structure. For example, R_{42} = {({1}, {5}, φ_{15}, 4, 2)}.

Proposition 1.5.13. Let L = (K, S, {F_{i}}_{i∈S}) be a rationally ramified selfsimilar structure with a relation set R.

(1) K ≠ \overline{V}_{0}.

(2) Set M = max_{x∈K} #(π^{−1}(x)). Suppose that π(ω) = π(τ) and ω ≠ τ. Then there exist Ω_{1}, …, Ω_{m} ∈ [R] and ω^{(1)}, …, ω^{(m+1)} ∈ Σ which satisfy the following conditions (AS1), (AS2) and (AS3):

(AS1) m + 1 ≤ max_{x∈K} #(π^{−1}(x)) (AS2) ω = ω^{(1)}, ω^{(m+1)} = τ and (ω^{(i)}, ω^{(i+1)}) is a corresponding pair with respect to Ω_{i} for any i = 1, …, m. (AS3) s(ω^{(i)}, τ) < s(ω^{(i+1)}, τ) for any i = 1, …, m − 2, where s(δ, ρ) = min{k − 1|δ_{k} ≠ ρ_{k}} for δ = δ_{1}δ_{2} … and ρ = ρ_{1}ρ_{2} . . ..

Remark. Under the assumptions of the above proposition, let ω^{(1)}, …, ω^{(m+1)} and Ω_{1}, …, Ω_{m} satisfy (AS1), (AS2) and (AS3). Set m_{n} = s(τ, ω^{(n)}). If Ω_{n} = (X_{n}, Y_{n}, φ_{n}, x(n), y(n)), then the first symbols of x(i) and y(i) are ω_{m_{n}+1}^{(n)} and τ_{m_{n}+1} respectively. Furthermore, ω^{(n)} = τ_{1} … τ_{m_{n}}x(n)x_{1}x_{2} … for some x_{1}x_{2} ⋯ ∈ Σ[X] and ω^{(n+1)} = τ_{1} … τ_{m_{n}}y(n)φ_{n}(x_{1})φ_{n}(x_{2}) . . ..

Proof. (1) Since C_{L} = ∪_{(X,Y,φ,x,y)∈R}Σ_{x}[X], the post critical set P_{L} is a finite union of Σ_{w}[X]’s for some w ∈ W_{∗} and some X where (X, Y, φ, x, y) ∈ R. Now V_{0} is a finite union of F_{w}(K[X])’s. Lemma 1.5.2 shows that V_{0} = \overline{V}_{0}. By [28, Corollaries 1.4.8 and 1.4.9], L is minimal. Hence, [28, Theorem 1.3.8] implies that K ≠ V_{0} = \overline{V}_{0}.

(2) Define ω^{(1)}, ω^{(2)}, … and Ω_{1}, Ω_{2}, … inductively as follows. Set ω^{(1)} = ω. Suppose we have ω^{(1)}, …, ω^{(n)} and Ω_{1}, …, Ω_{n−1}. If ω^{(n)} = τ, then we set m = n+1 and finish the construction. If ω^{(n)} ≠ τ, then set k = s(ω^{(n)}, τ)+1, i = ω_{k}^{(n)} and j = τ_{k}. By (1.5.1), we may choose Ω_{n} = (X, Y, φ, x, y) ∈ R_{ij} such that σ^{k}(ω^{(n)}) ∈ Σ_{x}[X]. Define ω^{(n+1)} = τ_{1} … τ_{k}φ_{∗}(σ^{k}(ω^{(n)}). Then ω_{k+1}^{(n+1)} = τ_{k+1}. Hence s(ω^{(n+1}), τ) > k. As far as this construction continues, ω^{(1)}, ω^{(2)}, …, ω^{(n)} and τ are mutually different elements. Therefore, n + 1 will not exceed max_{x∈K} #(π^{−1}(x)). □

The next two lemmas describe fine structures of intersections of two copies K_{w} and K_{v} for a rationally ramified self-similar set. They are technically useful in getting results in the following sections.

Lemma 1.5.14. Let L be rationally ramified and let (X, Y, φ, x, y) ∈ R_{L}. Define φ̂ : Σ[X] → Σ[Y] by φ̂(x_{1}x_{2} …) = φ(x_{1})φ(x_{2}) . . .. Then there exists a unique homeomorphism φ̃ : K[X] → K[Y] that satisfies φ̃ ∘ π = π ∘ φ̂.

Proof. Fix p ∈ K[X]. If there exist x_{1}x_{2} … ∈ Σ(X) and x′_{1}x′_{2} … ∈ Σ(X) such that π(x_{1}x_{2} …) = π(x′_{1}x′_{2} …) = p, then π(yφ̂(x_{1}x_{2} …)) = π(xx_{1}x_{2} …) = π(xx′_{1}x′_{2} …) = π(yφ̂(x′_{1}x′_{2} …)). Therefore, π(φ̂(x_{1}x_{2} …)) = π(φ̂(x′_{1}x′_{2} …)). Hence for any p ∈ K[X], π(φ̂(π^{−1}(x))) contains only one point. Define φ̃(p) as this one point. Then by a routine argument, φ̃ : K[X] → K[Y] is continuous. Exchanging X and Y, we obtain the inverse of φ̃. Hence φ̃ is a homeomorphism. □

Definition 1.5.15. Let X be a finite subset of W_{#} and let x ∈ W_{∗}. For each w ∈ W_{∗}, define

A_{X,x}(w) = {(z, x_{0}, x_{1}, …, x_{m})|m ≥ 0, z ∈ W_{∗},

x_{0} = x, x_{1}, …, x_{m} ∈ X, zx_{0}x_{1} … x_{m} ≤ w < zx_{0}x_{1} … x_{m−1}}.

Lemma 1.5.16. Let L be rationally ramified and let Ω = (X, Y, φ, x, y) ∈ R_{L}. Suppose w = w_{1} … w_{m}, v = v_{1} … v_{n} ∈ W_{#} and Σ_{w} ∩ Σ_{v} = ∅. Set z_{∗} = w_{1} … w_{N}, where N = inf{i|w_{i} = v_{i}} − 1. Then, there exist a corresponding pair with respect to Ω in Σ_{w} × Σ_{v} if and only if there exist (z_{∗}, x, x_{1}, …, x_{m}) ∈ A_{X,x}(w) and (z_{∗}, y, y_{1}, …, y_{n}) ∈ A_{Y,y}(v) such that y_{i} = φ(x_{i}) for i = 1, …, min (m, n). Moreover, let x_{m} = x_{m}^{1}x_{m}^{2} where w = zxx_{1}…x_{m−1}x_{m}^{1} and let y_{n} = y_{n}^{1}y_{n}^{2} where v = zyy_{1}…y_{n−1}y_{n}^{1}. Suppose that m ≥ n and define y_{n+1}, …, y_{m} by y_{i} = φ(x_{i}). Then K_{wx_{m}^{2}}[X] = K_{vy_{n}^{2}y_{n+1}...y_{m}}[Y] ⊆ K_{w} ∩ K_{v} and (F_{v})^{−1}∘F_{w}|_{K_{x^{2}_{m}} [X]} = F_{y_{n}^{2}y_{n+1}...y_{m}} ∘ φ̃ ∘ (F_{x_{m}^{2}})^{−1}, where φ̃ is the homeomorphism between K[X] and K[Y] introduced in Lemma 1.5.14. See the following commutative diagram, where y_{∗} = y_{n}^{2}y_{n+1} … y_{m}.

K[X] ⟶^{F_{x^{2}_{m}}} K_{x_{m}^{2}}[X] ⟶^{F_{w}} K_{wx_{m}^{2}}[X]
φ̃↓  ↓  ∥
K[Y] ⟶_{F_{y∗}} K_{y_{∗}}[X] ⟶_{F_{v}} K_{vy_{∗}}[Y]

1.6. Rationally ramified self-similar sets 2

Continued from the last section, we will focus on rationally ramified self-similar structures. In this class, there are useful criteria for a scale being locally finite and self-similar scales being gentle with respect to each other. As in the previous sections, L = (K, S, {F_{i}}_{i∈S}) is a self-similar structure.

Theorem 1.6.1. Let L be rationally ramified and let R be its relation set. Define R_{2} = {(X, Y, φ, x, y) ∈ R|#(X) ≥ 2}. Then an elliptic scale 𝒮 on Σ is locally finite with respect to L if and only if there exist c_{1}, c_{2} > 0 such that

(1.6.1)  c_{1}l(zxx_{1} … x_{m}) ≤ l(zyφ(x_{1}) … φ(x_{m})) ≤ c_{2}l(zxx_{1} … x_{m})

for any (X, Y, φ, x, y) ∈ R_{2}, any x_{1} … x_{m} ∈ W_{∗}(X) and any z ∈ W_{∗}, where l is the gauge function of 𝒮. In particular, for a ∈ (0, 1)^{S}, a self-similar scale 𝒮(a) on Σ is locally finite with respect to L if and only if a_{w} = a_{φ(w)} for any (X, Y, φ, x, y) ∈ R_{2} and any w ∈ X.

Corollary 1.6.2. Let L be rationally ramified. Assume that 𝒮_{1} · 𝒮_{2} is locally finite with respect to L for elliptic scales 𝒮_{1} and 𝒮_{2} on Σ. Then 𝒮_{1} is locally finite with respect to L if and only if 𝒮_{2} is locally finite with respect to L.

To prove Theorem 1.6.1, we need the following lemma.

Lemma 1.6.3. Let X be a nonempty independent finite subset of W_{#} and let x ∈ W_{#}. Assume that O_{Σ[X],x}(ω) < +∞ for any ω ∈ Σ. Then sup_{w∈W_{#}} #(A_{X,x}(w)) < +∞.

Proof. By Theorem 1.5.6, we may choose k so that Q_{k}(X) = ∅. Fix z_{∗} ∈ W_{∗}. Then (z_{∗}, x, x_{1}, …, x_{m}) ∈ A_{X,x}(w) is uniquely determined except for x_{m−k}, …, x_{m}. Therefore

(1.6.2)  sup_{w,z_{∗}} #{(z_{∗}, x, x_{1}, …, x_{m}) ∈ A_{X,x}(w)} < +∞.

Now define

N_{X,x}(w) = #{z|(z, x, x_{1}, …, x_{m}) ∈ A_{X,x}(w) for some (x_{1}, …, x_{m})}

Lemma 1.5.8 implies that sup_{ω∈Σ} O_{Σ[X],x}(ω) < +∞. Denote the value of this supremum by N. Suppose that N_{X,x}(w) > k(N + 1), where k = max_{w∈X} |w|. For (z, x, x_{1}, …, x_{m}) ∈ A_{X,x}(w), |w| − k ≤ |zxx_{1} … x_{m}| ≤ |w| − 1. Therefore, for some l ∈ {|w| − k, …, |w| − 1}, there exists {(z^{(i)}, x, x_{1}^{(i)}, …, x_{m_{i}}^{(i)})}_{i=1}^{N+1} ⊂ A_{X,x}(w) such that z^{(i)} ≠ z^{(j)} for any i ≠ j and z^{(i)}xx_{1}^{(i)} … x_{m_{i}}^{(i)} = w_{1} … w_{l}, where w = w_{1} … w_{|w|}. Set ω = w_{1} … w_{l}(x_{∗})^{∞}, where x_{∗} ∈ X. Then, σ^{|z^{(i)}|}ω ∈ Σ_{x}[X] for any i = 1, …, N+1. This contradicts to the definition of N. Hence, N_{X,x}(w) ≤ k(N+1) for any w ∈ W_{∗}. Combining this with (1.6.2), we have the desired estimate. □

Definition 1.6.4. Let L be a self-similar structure and let R ⊆ R_{L}. For a scale {Λ_{s}}_{s∈(0,1]}, we define

Λ_{s,w}^{R} = {v|v ∈ Λ_{s}, there exists an corresponding pair

with respect to some Ω ∈ [R] in Σ_{w} × Σ_{v}}

for any s ∈ (0, 1] and any w ∈ Λ_{s}.

Lemma 1.6.5. Let R be a relation set of a rationally ramified self-similar structure L and let 𝒮 = {Λ_{s}}_{s∈(0,1]} be a scale on L.

(1) 𝒮 is locally finite if and only if there exists C > 0 such that #(Λ_{s,w}^{R}) ≤ C for any s ∈ (0, 1] and any w ∈ Λ_{s}.

(2) Let ψ : W_{∗} → [0, +∞). Then, ψ is gentle with respect to 𝒮 if and only if there exists C′ > 0 such that f(w) ≤ C′f(v) for any s ∈ (0, 1], any w ∈ Λ_{s} and any v ∈ Λ_{s,w}^{R}.

Proof. Let M = max_{x∈K} #(π^{−1}(x)). If v ∈ Λ_{s,w}, then there exists p ∈ K_{w} ∩ K_{v}. Choose ω and τ ∈ π^{−1}(p) so that ω ∈ Σ_{w} and τ ∈ Σ_{v}. By Proposition 1.5.13- (2), we have ω^{(1)}, …, ω^{(m+1)} ∈ Σ and Ω_{1}, …, Ω_{m} ∈ [R] with (AS1), (AS2) and (AS3). Hence, if W_{s,w} = Λ_{s,w}^{R} ∪ {w},

(1.6.3)  Λ_{s,w} ⊆ ⋃_{w^{(1)}∈W_{s,w}} ⋃_{w^{(2)}∈W_{s,w^{(1)}}} … ⋃_{w^{(M−1)}∈W_{s,w^{(M−2)}}} W_{s,w^{M−1}}.

Now if #(Λ_{s,w}^{R}) ≤ C for any s and w, then (1.6.3) implies that #(Λ_{s,w}) ≤ (C +1)^{M}. Hence we have (1). Next suppose that f(w) ≤ C′f(v) for any w ∈ Λ_{s} and any v ∈ Λ_{s,w}^{R}. Then by (1.6.3), f(w) ≤ (C′)^{M−1}f(v) for any w ∈ Λ_{s} and any v ∈ Λ_{s,w}. This shows (2). □

Proof of Theorem 1.6.1. Note that Ω ∈ [R] is a finite composition of subrelations of relations in R and the the number of composed sub-relations is uniformly bounded. Therefore, we may assume that R = [R] without loss of generality. Let 𝒮 = {Λ_{s}}_{0<s≤1}. Since 𝒮 is elliptic, there exist δ_{1}, δ_{2} ∈ (0, 1) and c > 0 such that (δ_{1})^{|v|}l(w) ≤ l(wv) ≤ c(δ_{2})^{|v|}l(w). Define

M = max{|w||w ∈ X or w ∈ Y for some (X, Y, φ, x, y) ∈ R_{2}}.

By Theorem 1.5.6, we may choose n ≥ 1 so that Q_{n}(X) = ∅ for any X with (X, Y, φ, x, y) ∈ R_{2}. Assume that there exist (X, Y, φ, x, y) ∈ R_{2}, x_{1} … x_{m} ∈ W_{∗}(X) and z ∈ W_{∗} such that l(zxx_{1} … x_{m})(δ_{1})^{M(k+n)} > l(zyy_{1} … y_{m}), where y_{j} = φ(x_{j}) for j = 1, …, m. Set s = l(zyy_{1} … y_{m}). Then there exists v ∈ W_{∗} such that v ≥ zyy_{1} … y_{m} and v ∈ Λ_{s}. Since l(zxx_{1} … x_{m+k+n}) > s for any (x_{m+1}, …, x_{m+k+n}) ∈ X^{k+n}, Lemma 1.5.16 implies that there exists w ∈ W_{∗} such that w ≤ zxx_{1} … x_{m+k+n} and w ∈ Λ_{s,v}. Since Q_{n}(X) = 0, the set

{w|x_{m+1}, …, x_{m+k+n} ∈ X, w ≤ zxx_{1} … x_{m+k+n}, w ∈ Λ_{s,v}}

contains 2^{k} elements at least. Hence #(Λ_{s,v}) ≥ 2^{k}. Therefore, if 𝒮 is locally finite with respect to L, then we have (1.6.1) by Lemma 1.3.6.

Next we assume (1.6.1). Let w ∈ Λ_{s} and let Ω = (X, Y, φ, x, y) ∈ R_{2}. For γ = (z, x, x_{1}, …, x_{m}) ∈ A_{X,x}(w), we define

B(γ) = {(v, z, y, y_{1}, …, y_{n})|v ∈ Λ_{s}, (z, y, y_{1}, …, y_{n}) ∈ A_{Y,y}(v),

x_{i} = φ(y_{i}) for i = 1, …, min (m, n)}.

By Lemma 1.5.16, for v ∈ Λ_{s,w}^{R}, there exist γ ∈ A_{X,x}(w) and (v, z, y, y_{1}, …, y_{n}) ∈ B(γ). If #(X) = 1, then it is immediate to see #(B(γ)) = 1. Suppose #(X) ≥ 2. Let γ_{∗} = (v, y, y_{1}, …, y_{n}) ∈ B(γ). Since both w and v belongs to Λ_{s},

c_{3}l(zxx_{1} … x_{m}) ≤ l(zyy_{1} … y_{n}) ≤ c_{4}l(zxx_{1} … x_{m}),

where c_{3} and c_{4} are positive constants which are independent of s, w, Ω, γ and γ_{∗}. If n ≥ m, then l(zyy_{1} … y_{n}) ≤ cδ_{2}^{k}l(zyy_{1} … y_{m}), where k = |n − m|. Hence c_{3}l(zxx_{1} … x_{m}) ≤ c(δ_{2})^{k}l(zyy_{1} … y_{m}). By (1.6.1), |n−m| is bounded by a constant which is independent of s, w, Ω, γ and γ_{∗}. (Note that the above discussion is valid even if n < m; we only need to exchange γ and γ_{∗} and do the same argument.) Therefore, #(B(γ)) is uniformly bounded with respect to w, γ. This fact with Lemma 1.6.3 implies that #(Λ_{s,w}^{R}) is uniformly bounded with respect to s and w. By Lemma 1.6.5-(1), 𝒮 is locally finite with respect to L.

Finally if 𝒮 = 𝒮(a), then it is straightforward to show that (1.6.1) is equivalent to that a_{w} = a_{φ(w)} for any (X, Y, φ, x, y) ∈ R_{2} and any w ∈ X. □

For the gentleness of self-similar scales, we have the following result.

Theorem 1.6.6. Let L = (K, 𝒮, {F_{i}}_{i∈S}) be rationally ramified and let R be a relations set of L. For a = (a_{i})_{i∈S}, b = (b_{i})_{i∈S} ∈ (0, 1)^{S}, 𝒮(a) ∼_{GE} 𝒮(b) if and only

if, for any (X, Y, φ, x, y) ∈ R, either (R1) or (R2) below is satisfied: (R1) a_{w} = a_{φ(w)} and b_{w} = b_{φ(w)} for any w ∈ X. (R2) There exists δ > 0 such that

δ = \begin{matrix}log & a_{w} \\ log & b_{w} \end{matrix} = \begin{matrix}log & a_{φ(w)} \\ log & b_{φ(w)} \end{matrix}

for any w ∈ X.

Proof. We may assume that R = [R] without loss of generality. First assume that every Ω ∈ R satisfies (R1) or (R2). Suppose that v ∈ Λ_{s,w}(a)^{R}. Then we find a corresponding pair (ω, τ) with respect to some (X, Y, φ, x, y) ∈ R satisfying ω ∈ Σ_{w}, τ ∈ Σ_{v} and π(ω) = π(τ). Now, let ω = zxx_{1}x_{2} … and let τ = zyy_{1}y_{2} …, where z ∈ W_{∗}, x_{1}, x_{2}, … ∈ X and y_{i} = φ(x_{i}) for any i. Then we obtain that w = zxx_{1} … x_{n}x′ and v = zyy_{1} … y_{m}y′, where x_{n+1} < x′ and y_{m+1} < y′. Assume that (R2) holds. Then, b_{y_{i}} = (a_{x_{i}})^{δ} for any i. Now

\frac{b_{w}}{b_{v}} = \frac{b_{x}b_{x′}a_{w}^{δ}(a_{x}a_{x′})^{−δ}}{b_{y}b_{y′}a_{v}^{δ}(a_{y}a_{y′})^{−δ}} = (\frac{a_{w}}{a_{v}} )^{δ}(\frac{a_{y}a_{y′}}{a_{x}a_{x′}} )^{δ} \frac{b_{x}b_{x′}}{b_{y}b_{y′}}

Note that a_{w}/a_{v} is bounded (from above and below) by Lemma 1.1.13 Also since R and X is a finite set, a_{x}, a_{x′}, b_{y}, b_{y′} is uniformly bounded. Therefore, b_{w}/b_{v} is uniformly bounded. If (R1) is satisfied, then

\frac{a_{w}}{a_{v}} = \frac{a_{x}a_{x′}}{a_{y}a_{y′}} a_{x_{m+1}} … a_{x_{n}},

where we assume that n ≥ m. Since a_{w}/a_{v} is uniformly bounded, it follows that |m − n| is uniformly bounded from above. Hence

\frac{b_{w}}{b_{v}} = \frac{b_{x}b_{x′}}{b_{y}b_{y′}} b_{x_{m+1}} … b_{x_{n}}

is uniformly bounded (from above and below). Hence Lemma 1.6.5-(2) implies that

𝒮(a) ∼_{GE} 𝒮(b).

Conversely assume that 𝒮(a) ∼_{GE} 𝒮(b). Let (X, Y, φ, x, y) ∈ R and let w ∈ X.

Since π(x(w)^{∞}) = π(y(φ(w))^{∞}), Lemma 1.4.6 implies

(1.6.4)  \begin{matrix}log & a_{w} \\ log & b_{w} \end{matrix} = \begin{matrix}log & a_{φ(w)} \\ log & b_{φ(w)} \end{matrix}

We write δ_{w} = log b_{w}/ log a_{w}. For x_{1} ≠ x_{2} ∈ X, write y_{i} = φ(x_{i}) for i = 1, 2. Note that π(x(x_{1}x_{2})^{∞}) = π(y(y_{1}y_{2})^{∞}). Hence by Lemma 1.4.6, we obtain (1.6.4) with w = x_{1}x_{2}. Combining the three equations (1.6.4) with w = x_{1}, x_{2} and x_{1}x_{2}, we obtain either

(1.6.5)  δ_{x_{1}} = δ_{x_{2}},

or

(1.6.6)  \begin{matrix}log & a_{x_{1}} \\ log & a_{y_{1}} \end{matrix} = \begin{matrix}log & b_{x_{1}} \\ log & b_{y_{1}} \end{matrix} = \begin{matrix}log & a_{x_{2}} \\ log & a_{y_{2}} \end{matrix} = \begin{matrix}log & b_{x_{2}} \\ log & b_{y_{2}} \end{matrix}

is satisfied. Suppose that (1.6.5) does not hold. Then we have (1.6.6). Write p = log a_{y_{1}}/ log a_{x_{1}}. Then a_{y_{i}} = (a_{x_{i}})^{p} and b_{y_{i}} = (b_{x_{i}})^{p} for i = 1, 2. Without loss of generality, we may assume that 0 < p ≤ 1. (If not, exchange X and Y.) Suppose that p ≠ 1. Set x(m) = x(x_{1})^{m} for any m ≥ 1. Define s_{m} = a_{x(m)} = a_{x}(a_{x_{1}})^{m}. As 0 < p < 1, for sufficiently large m, there exists a unique n(m) ∈ ℕ such that

(1.6.7)  a_{y}(a_{y_{1}})^{m}(a_{y_{2}})^{n(m)−1} > s_{m} ≥ a_{y}(a_{y_{1}})^{m}(a_{y_{2}})^{n(m)}.

Then y(m) = y(y_{1})^{m}(y_{2})^{n(m)} ∈ Λ_{s_{m}}(a). Since a_{y_{i}} = (a_{x_{i}})^{p}, (1.6.7) implies that

(1.6.8)  n(m) − 1 ≤ \frac{log a_{x} − log a_{y}}{log a_{x_{2}}} + m\frac{(1 − p)}{p} \begin{matrix}log & a_{x_{1}} \\ log & a_{x_{2}} \end{matrix} ≤ n(m).

Note that x(m), y(m) ∈ Λ_{s_{m}}(a). Hence b_{x(m)}/b_{y(m)} is uniformly bounded from below and above with respect to m because 𝒮(a) ∼_{GE} 𝒮(b). Now b_{x(m)} = b_{x}(a_{x_{1}})^{mδ_{x_{1}}} .

Using Lemma 1.4.6, we obtain (1.6.4) with w = (x_{1})^{m}(x_{2})^{n(m)}. Therefore, if δ_{m} = δ_{(x_{1})^{m}(x_{2})^{n(m)}}, then b_{y(m)} = b_{y}((a_{y_{1}})^{m}(a_{y_{2}})^{n(m)})^{δ_{m}}. Hence,

\frac{b_{x(m)}}{b_{y(m)}} = \frac{b_{x}}{b_{y}} (\frac{a_{y}}{a_{x}} )^{δ_{m}}(\frac{a_{x(m)}}{a_{y(m)}} )^{δ_{m}}(a_{x_{1}})^{m(δ_{x_{1}} −δ_{m})}.

As min(δ_{x_{1}}, δ_{x_{2}}) ≤ δ_{m} ≤ max(δ_{x_{1}}, δ_{x_{2}}), the first three factors in the above equality is uniformly bounded from above and below with respect to m. Therefore, so is the fourth factor (a_{x_{1}})^{m(δ_{x_{1}} −δ_{m})}. On the other hand, by (1.6.8),

lim_{m→∞}(δ_{x_{1}} − δ_{m}) = \begin{matrix}log & b_{x_{1}} \\ log & a_{x_{1}} \end{matrix} − \frac{log b_{x_{1}} + A log b_{x_{2}}}{log a_{x_{1}} + A log a_{x_{2}}} ,

where A = \frac{(1 − p)}{p} \begin{matrix}log & a_{x_{1}} \\ log & a_{x_{2}} \end{matrix} . Now since 0 < p < 1 and δ_{x_{1}} ≠ δ_{x_{2}}, the value of the

above limit is not zero. Therefore, (a_{x_{1}})^{m(δ_{x_{1}} −δ_{m})} is not uniformly bounded from above and below with respect to m. This contradiction implies that p = 1. Thus it follows that, for any x_{1} ≠ x_{2} ∈ X, (1.6.5) holds or a_{x_{i}} = a_{y_{i}} and b_{x_{i}} = b_{y_{i}} for i = 1, 2. So if (R1) is not satisfied (i.e. there exists some w ∈ X such that a_{w} ≠ a_{φ(w)} or b_{w} ≠ b_{φ(w)}), then δ_{w} = δ_{w′} for any w′ ∈ X with w ≠ w′. This implies (R2). Thus we have the desired conclusion. □

Combining Theorems 1.6.1 and 1.6.6, we can show that the number of equivalence classes of locally finite self-similar scales under ∼_{GE} is 0, 1 or +∞ as follows.

Theorem 1.6.7. Let L = (K, S, {F_{i}}_{i∈S}) be a rationally ramified self-similar structure and let R be its relation set. For any w = w_{1} … w_{m} ∈ W_{∗}, we define f_{w} ∈ ℓ(S) by f_{w}(i) = #({k|w_{k} = i}) for any i ∈ S. Define R_{1} = {(X, Y, φ, x, y) ∈ R|#(X) = 1} and R_{2} = {(X, Y, φ, x, y) ∈ R|#(X) ≥ 2}. Also let U be the subspace of ℓ(S) generated by {f_{w} − f_{φ(w)}|(X, Y, φ, x, y) ∈ R_{2}, w ∈ X}. (If R_{2} = ∅, then U is thought of as {0}.)

(1) There exists a self-similar scale on Σ which is locally finite with respect to L if and only if U ∩ [0, +∞)^{S} = {0}.

(2) Assume that U ∩ [0, +∞)^{S} = {0}. For Ω = (X, Y, φ, x, y) ∈ R_{1}, we use U_{Ω} to denote the subspace of ℓ(S) generated by {f_{w}, f_{φ(w)}}, where w ∈ X. Also define

𝔖_{LF}(Σ, L) = {𝒮|𝒮 ∈ 𝔖(Σ), 𝒮 is locally finite with resepct to L}.

If for any Ω ∈ R_{1} with dim U_{Ω} = 2, then #(𝔖_{LF}(Σ, L)/∼_{GE}) = 1. In other words, all self-similar scales which are locally finite with respect to L are gentle each other if dim (U ∩ U_{Ω}) = 1. If dim (U ∩ U_{Ω}) = 0 for some Ω ∈ R_{1} with dim U_{Ω} = 2, then

#(𝔖_{LF}(Σ, L)/∼_{GE}) = +∞.

Remark. Let Ω = (X, Y, φ, x, y) ∈ R_{1} and let X = {w}. Then both f_{w} and f_{φ(w)} belong to [0, +∞)^{S}. Therefore, if U ∩ [0, ∞)^{S} = {0} and dim (U ∩ U_{Ω}) > 0, then dim (U ∩ U_{Ω}) = 1.

The following lemma is a version of Stiemke’s Theorem (or Minkowski-Frakas’s Theorem), which is a well-known result in convex analysis. See [39] or [37] for example.

Lemma 1.6.8. Let X be a finite set and U be a subspace of ℓ(X). Then U ∩ [0, +∞)^{X} = {0} if and only if U^{⊥} ∩ (0, +∞)^{X} ≠ ∅, where U^{⊥} is the orthogonal complement with respect to the inner product (·, ·)_{X}.

Lemma 1.6.9. Assume that U ∩ [0, +∞)^{S} = {0}. Let Ω = ({w}, {v}, φ, x, y) ∈

R_{1}. Define Φ_{Ω} : U^{⊥} → ℝ^{2} by Φ_{Ω}(p) = (\begin{matrix}(f_{w}, p)_{S} \\ (f_{v}, p)_{S} \end{matrix}). Then dim Φ_{Ω}(U^{⊥}) = 1 or 2.

Moreover, dim Φ_{Ω}(U^{⊥}) = 1 if and only if dim U_{Ω} = 1 or dim (U ∩ U_{Ω}) = 1.

Proof. By Lemma 1.6.9, we have U^{⊥} ∩ (0, +∞)^{S} ≠ ∅. Hence Φ_{Ω}(U^{⊥}) ≠ {0}. Hence dim Φ_{Ω}(U^{⊥}) > 0. Since U^{⊥} ∩ (0, +∞)^{S} ≠ ∅, it follows that dim Φ_{Ω}(U^{⊥}) = 1 if and only if there exist α > 0 and β > 0 such that αf_{w} − βf_{v} ∈ (U^{⊥})^{⊥} = U. αf_{w} −βf_{v} = 0 if and only if dim U_{Ω} = 1. Also αf_{w} −βf_{v} ≠ 0 if and only if U ∩U_{Ω} ≠ 0. By the remark after Theorem 1.6.7, this is equivalent to dim (U ∩ U_{Ω}) = 1. □

Proof of Theorem 1.6.7. Let 𝒮 = 𝒮(a), where a = (a_{i})_{i∈S} ∈ (0, 1)^{S}. Set p_{i} = log a_{i} for i ∈ S and write p = (p_{i})_{i∈S}. Note that p ∈ (−∞, 0)^{S}. By Theorem 1.6.1, 𝒮 is locally finite if and only if (f_{w}−f_{φ(w)}, p)_{S} = 0 for any (X, Y, φ, x, y) ∈ R_{2} and any w ∈ X. This is equivalent to that p ∈ U^{⊥}. Therefore, there exists a self-similar scale which is locally finite if and only if U^{⊥} ∩ (−∞, 0)^{S} ≠ ∅. Since U is a linear subspace, U^{⊥} ∩ (−∞, 0)^{S} ≠ ∅ if and only if U^{⊥} ∩ (0, +∞)^{S} ≠ ∅. Now Lemma 1.6.8 immediately implies (1).

Next we assume that U ∩ [0, +∞)^{S} = {0}. Let 𝒮(a) and 𝒮(b) be locally finite. By Theorem 1.6.1, the condition (R1) is satisfied for any (X, Y, φ, x, y) ∈ R_{2}. If dim (U ∩ U_{Ω}) = 1 for any Ω ∈ R_{1} with dim U_{Ω} = 2, then Lemma 1.6.9 shows that dim Φ_{Ω}(U^{⊥}) = 1 for any Ω ∈ R_{1}. This immediately implies that (R2) holds for any Ω ∈ R_{1}. Thus we see 𝒮(a) ∼_{GE} 𝒮(b). Now assume that dim (U ∩ U_{Ω}) = 0 for some Ω ∈ R_{1} with dim U_{Ω} = 2. Then by Lemma 1.6.9 implies that dim_{Φ_{Ω}(U^{⊥})} = 2. Then for any q ∈ (−∞, 0)^{2}, there exists a = (a_{i})_{i∈S} such that log a ∈ U^{⊥} ∩ (−∞, 0)^{S} and Φ_{Ω}(log a) = q, where log a = (log a_{i})_{i∈S} ∈ ℓ(S). Therefore there is no constraint on the ratio between log a_{w} and log a_{v}, where Ω = ({w}, {v}, φ, x, y). Hence #(𝔖_{LF}(Σ, L)/∼_{GE}) = +∞. □

Corollary 1.6.10. Let L be rationally ramified and let R_{1} and R_{2} be the same as in Theorem 1.6.7. If R_{1} = ∅ and U ∩ [0, +∞)^{S} = {0}, then #(𝔖_{LF}(Σ, L)/∼_{GE}) = 1.

In the case of post critically finite self-similar structures, the above results are easy to verify as follows.

Definition 1.6.11. A self-similar structure L = (K, S, {F_{i}}_{i∈S}) is called post critically finite (p. c. f. for short) if and only if the post critical set P of L is a finite set.

Proposition 1.6.12. L = (K, S, {F_{i}}_{i∈S}) is post critically finite if and only if L is rationally ramified with a relation set R which satisfies R = R_{1}. Moreover, if L is post critically finite, then any scale 𝒮 of Σ is locally finite with respect to L.

Corollary 1.6.13. Suppose that L is post critically finite with a relation set R. Let

R = {({w(j)}, {v(j)}, φ_{j}, x(j), y(j))|j = 1, …, k, w(j), v(j), x(j), y(j) ∈ W_{#}},

where φ_{j}(w(j)) = v(j).

(1) For a = (a_{i})_{i∈S}, b = (b_{i})_{i∈S} ∈ (0, 1)^{S}, 𝒮(α) ∼_{GE} 𝒮(β) if and only if

(1.6.9)  \begin{matrix}log & a_{w(j)} \\ log & b_{w(j)} \end{matrix} = \begin{matrix}log & a_{v(j)} \\ log & b_{v(j)} \end{matrix}

for all i = 1, …, k.

(2) Let a = (a_{i})_{i∈S} ∈ (0, 1)^{S}. A self-similar measure μ with weight (μ_{i})_{i∈S} has volume doubling property with respect to 𝒮(a) if and only if (1.6.9) with b = (μ_{i})_{i∈S} holds for all j = 1, …, k.

1.7. Examples

In this section, we will apply our results in the previous sections to several examples.

Example 1.7.1 (Sierpinski gasket). Let (K, S, {F_{i}}_{i∈S}) be the same as in Example 1.5.11. By Corollary 1.6.13, for a = (a_{i})_{i∈S}, b = (b_{i})_{i∈S} ∈ (0, 1)^{S}, 𝒮(a) ∼_{GE} 𝒮(b) if and only if there exists δ > 0 such that δ = \begin{matrix}log & b_{i} \\ log & a_{i} \end{matrix} for any

i ∈ S. Hence {𝒮|𝒮 is a self-similar scale and 𝒮 ∼_{GE} 𝒮(a)} = {𝒮(a^{δ})|δ > 0}, where

a^{δ} = {(a_{i})^{δ}}_{i∈S}. Also a self-similar measure μ with weight (μ_{i})_{i∈S} has volume doubling property with respect to 𝒮(a) if and only if μ_{i} = (a_{i})^{d}, where d is the unique constant that satisfies ∑_{i∈S}(a_{i})^{d} = 1.

Define M_{VD}^{S}(L, 𝒮) = {(μ)_{i∈S}|the self-similar measure with weight (μ_{i})_{i∈S}

has volume doubling property with respect to 𝒮}.

We always identify M_{VD}^{S} as the collections of self-similar measures with volume doubling property with respect to 𝒮. For the Sierpinski gasket, M_{VD}^{S}(L, 𝒮) consists of only one self-similar measure. In general, however, the collection of self-similar measures with volume doubling property may have richer structure. In fact, even for the Sierpinski gasket, this is the case if we change the self-similar structure.

Example 1.7.2. L = (K, S, {F_{i}}_{i∈S}) is the same as in Example 1.7.1. Define L_{2} = (K, W_{2}, {F_{w}}_{w∈W_{2}}). Then L_{2} is a p. c. f. self-similar structure with P_{L_{2}} = {(ii)^{∞}|i ∈ S}. If (X, Y, φ, x, y) belongs to the relation set of L_{2}, then X = {ii}, Y = {jj} and φ(ii) = jj for some i ≠ j ∈ S. Let a = (a_{w})_{w∈W_{2}} ∈ (0, 1)^{W_{2}} and consider 𝒮(a), the self-similar scale on Σ(W_{2}) with weight a. Also let μ be a self-similar measure with respect to L_{2} with weight (μ_{w})_{w∈W_{2}}. Then a self-similar measure μ with weight (μ_{ij})_{ij∈W_{2}} has the volume doubling property with respect to a selfsimilar scale (a_{ij})_{ij∈W_{2}} if and only if there exists δ > 0 such that μ_{ii} = (a_{ii})^{δ} for any i ∈ S. In particular, if i ≠ j, we may choose any value for μ_{ij} as long as ∑_{w∈S} μ_{w} = 1 and 0 < μ_{ij}. So M_{VD}^{S}(K, 𝒮) is an infinite set. This fact also shows that M_{VD}(L, 𝒮) is not trivial for any self-similar scale 𝒮 on Σ(S) because M_{VD}^{S}(L_{2}, 𝒮_{2}(a)) ⊂ M_{VD}(L, 𝒮(a)), where 𝒮_{2}(a) is the self-similar scale on Σ(W_{2}) with weight (a_{i}a_{j})_{ij∈W_{2}}.

Next we present two examples, unit square and the Sierpinski carpet, which are not post critically finite but rationally ramified.

Example 1.7.3 (Unit square). Let K be the unit square in ℝ^{2}, i.e. K = [0, 1]^{2} as in Section 0.2. We will identify ℝ^{2} with ℂ is the usual manner. Let p_{1} = 0, p_{2} = 1, p_{3} = 1+\sqrt{−1} and p_{4} = \sqrt{−1}. Define f_{i} : ℂ → ℂ by f_{i}(x) = (x−p_{i})/2+p_{i}. (\{f_{i}\}’s are the same as in Section 0.2.) Then, L = (K, S, {f_{i}}_{i∈S}), where S = {1, 2, 3, 4}, is a rationally ramified self-similar structure. To describe its relation set R, we define X_{1} = {1, 2}, Y_{1} = {4, 3}, φ_{1}(1) = 4, φ_{1}(2) = 3, X_{2} = {1, 4}, Y_{2} = {2, 3}, φ_{2}(1) = 2 and φ_{2}(4) = 3. As we explained in Section 0.2, where φ_{2} was denoted by ϕ, (X_{2}, Y_{2}, φ_{2}, 2, 1) is a relation. (See Figure 0.1.) In the same way, we have a relation set

R = {(X_{1}, Y_{1}, φ_{1}, 4, 1), (X_{1}, Y_{1}, φ_{1}, 3, 2), (X_{2}, Y_{2}, φ_{2}, 2, 1), (X_{2}, Y_{2}, φ_{2}, 3, 4)}.

Let a = (a_{i})_{i∈S} ∈ (0, 1)^{S} and let b = (b_{i})_{i∈S} ∈ (0, 1)^{S}. Then Theorem 1.6.6 implies that 𝒮(a) ∼_{GE} 𝒮(b) if and only if there exists δ > 0 such that log b_{i}/ log a_{i} = δ for

any i ∈ S. On the other hand, by Theorem 1.6.1, 𝒮(a) is locally finite with respect to L if and only if a_{1} = a_{2} = a_{3} = a_{4}. Hence,there is only one equivalence class in 𝔖(Σ)/∼_{GE} which consists of locally finite scales. Let μ be a self-similar measure on K and let a ∈ (0, 1)^{S}. Theorem 1.3.5 shows that μ has the volume doubling property with respect to 𝒮(a) if and only if a_{1} = a_{2} = a_{3} = a_{4} and μ is the restriction of the Lebesgue measure on K. So, we have only one choice of the volume doubling measure in this case. Note that if a_{i} = 1/2 for all i, U_{s}(x) is equivalent to the Euclidean ball, i.e. there exist c_{1} and c_{2} such that B_{c_{1}r}(x, d) ⊆ U_{r}(x) ⊆ B_{c_{2}r}(x, d) for any r ∈ [0, 1] and any x ∈ K, where d is the Euclidean distance. (In such a situation, the Euclidean distance is said to be adapted to the scale 𝒮(a). (See Section 2.3 for details.) This fact shows Theorem 0.2.1.

Changing the self-similar structure, however, we have richer structure as in the case of Sierpinski gaskets. Let S′ = {1, …, 9} and let {F_{i}}_{i∈S′} be the collection of contractions defined in Section 0.2. Then L′ = {K, S′, {F_{i}}_{i∈S′}} is a self-similar structure. Let

R = {({1, 8, 7}, {3, 4, 5}, ψ_{1}, x, y)|(x, y) = (2, 1), (2, 3), (9, 8), (4, 9), (6, 7), (5, 6)}

∪ {({1, 2, 3}, {7, 6, 5}, ψ_{2}, x, y)|(x, y) = (8, 1), (7, 8), (9, 2), (6, 9), (4, 3), (5, 4)},

where ψ_{1}(1) = 3, ψ_{1}(8) = 4, ψ_{1}(7) = 5, ψ_{2}(1) = 7, ψ_{2}(2) = 6, ψ_{2}(3) = 5. Then L′ is rationally ramified with a relation set R. By Theorem 1.6.1, a self-similar scale 𝒮(a) is locally finite with respect to L′ if and only if a_{1} = a_{3} = a_{5} = a_{7}, a_{2} = a_{6} and a_{4} = a_{8}. (In Section 0.2, a ratio {a_{i}}_{i∈S′} which satisfies this condition is said to be weakly symmetric.) Moreover, Corollary 1.6.10 implies that #(𝔖_{LF}(Σ, L)/∼_{GE}) = 1. Combining those results with Theorem 1.3.5, we see that a self-similar measure μ with weight {μ_{i}}_{i∈S′} has the volume doubling property with respect to a self-similar scale 𝒮(a) if and only if both {μ_{i}}_{i∈S′} and {a_{i}}_{i∈S′} are weakly symmetric. This fact essentially shows Theorem 0.2.3.

Example 1.7.4 (the Sierpinski Carpet). Let L be the self-similar structure associated with the Sierpinski carpet given in Example 1.5.12. Fix a = (a_{i})_{i∈S} ∈ (0, 1)^{S}. Using Theorem 1.6.6, we are going to determine if 𝒮(b) ∼_{GE} 𝒮(a) holds for

b = (b_{i})_{i∈S} or not. Define two conditions (SC1) and (SC2) as follows: (SC1) a_{1} = a_{7}, a_{2} = a_{6} and a_{3} = a_{5} (SC2) a_{1} = a_{3}, a_{8} = a_{4} and a_{7} = a_{5}

Then there are four cases: (A) Assume that both (SC1) and (SC2) are satisfied, i.e. a_{1} = a_{3} = a_{5} = a_{7}, a_{2} = a_{6} and a_{8} = a_{4}. See Figure 1.3. Then 𝒮(b) ∼_{GE} 𝒮(a) if and only if b_{1} = b_{3} =

b_{5} = b_{7}, b_{2} = b_{6} and b_{8} = b_{4}. So all self-similar scales on Σ with (SC1) and (SC2) are equivalent under ∼_{GE}. Theorem 1.6.1 implies that any scale in this class is locally finite. Moreover, by Corollary 1.6.10, this is the only one equivalence class in 𝔖(Σ)/∼_{GE} which consists of locally finite scales. Also in this case,

M_{VD}^{S}(L, 𝒮(a)) = {(μ_{i})_{i∈S}|μ_{1} = μ_{3} = μ_{5} = μ_{7}, μ_{2} = μ_{6}, μ_{4} = μ_{8}}.

(B) Assume that (SC1) holds but (SC2) does not. Then 𝒮(b) ∼_{GE} 𝒮(a) if and

only if b_{1} = b_{7} = (a_{1})^{δ}, b_{3} = b_{5} = (a_{3})^{δ}, b_{4} = (a_{4})^{δ}, b_{8} = (a_{8})^{δ} and b_{2} = b_{6} for some δ > 0. In this case, as we mentioned in (A), no scale is locally finite and M_{VD}(L, 𝒮(a)) = ∅. (C) Assume that (SC2) holds but (SC1) does not. Then 𝒮(b) ∼_{GE} 𝒮(a) if and only

if b_{1} = b_{3} = (a_{1})^{δ}, b_{5} = b_{7} = (a_{5})^{δ}, b_{2} = (a_{2})^{δ}, b_{6} = (a_{6})^{δ} and b_{8} = b_{4} for some

a  b  a c  c a  b  a  ^{a = a_{1} = a_{3} = a_{5} = a_{7}}^{b = a_{2} = a_{6}}^{c = a_{4} = a_{8}}

Figure 1.3. Case (A) of the Sierpinski carpet

δ > 0. In this case, no scale is locally finite and M_{VD}(L, 𝒮(a)) = ∅. (D) Assume that neither (SC1) nor (SC2) holds. Then 𝒮(b) ∼_{GE} 𝒮(a) if and only if

there exists δ > 0 such that b_{i} = (a_{i})^{δ} for any i ∈ S. In this case, no scale is locally finite and M_{VD}(L, 𝒮(a)) = ∅.

Next we introduce a class of self-similar sets which are modifications of the Sierpinski carpet. This class contains self-similar structures which are not rationally ramified.

Example 1.7.5 (Sierpinski cross). Let p_{1}, …, p_{8} be the same as in Examples 1.5.12 and1.7.4. For r ∈ [1/3, 1/2), define

F_{i}(z) = \begin{cases}r(z − p_{i}) + p_{i} & if i is odd, \\ (1 − 2r)(z − p_{i}) + p_{i} & if i is even. \end{cases}

The unique nonempty compact set K ⊆ ℝ^{2} satisfying K = ∪_{i=1}^{8}F_{i}(K) is called a Sierpinski cross. (Note that if r = 1/3, then K is the Sierpinski carpet.) Let S = {1, …, 8} and let L = (K, S, {F_{i}}_{i∈S}). In this case, L may (or may not) be rationally ramified. In fact, we have the following dichotomy.

Proposition 1.7.6. Let L be a Sierpinski cross. Then L is rationally ramified if and only if r is the unique positive solution of 1 − 2r = r^{m} for some m ∈ ℕ.

We will prove this proposition at the end of this section.

First we consider the rationally ramified cases. Assume that 1 − 2r = r^{m} for some m ∈ ℕ. Since L is the Sierpinski gasket for m = 1, we assume that m > 1. If X_{1}, Y_{1}, φ_{1}, X_{2}, Y_{2} and φ_{2} are the same as in Example 1.5.12, then the relation set Figure 1.4. Sierpinski cross: rationally ramified case r = \sqrt{2} − 1

Figure 1.5. Sierpinski cross: non rationally ramified case r = 2/5

R of L equals

{(X_{l}, Y_{l}, φ_{l}, i(j)^{m−1}, k), (X_{l}, Y_{l}, φ_{l}, k, j(i)^{m−1})

|(i, j, k, l) = (7, 1, 8, 1), (5, 3, 4, 1), (3, 1, 2, 2), (5, 7, 6, 2)}

Using Theorems 1.6.1 and 1.6.7, we see that 𝒮(a) is locally finite if and only if a_{1} = a_{3} = a_{5} = a_{7}, a_{2} = a_{6} and a_{4} = a_{8}. Obviously those scales are gentle each other and form an equivalence class of 𝔖(Σ)/∼_{GE}. Also a self-similar measure μ with

weight (μ_{i})_{i∈S} has volume doubling property with respect to those scales if and only if μ_{1} = μ_{3} = μ_{5} = μ_{7}, μ_{2} = μ_{6} and μ_{4} = μ_{8}.

Even if L is not rationally ramified, there exists at lease one self-similar scale on Σ that is locally finite with respect to L. Define c = (c_{i})_{i∈S} by c_{i} = r if i is odd, c_{i} = 1 − 2r if i is even. For any w ∈ W_{∗}, define ∂K_{w} as the topological boundary of F_{w}([0, 1]^{2}). (In fact, ∂K_{w} = F_{w}(V_{0}).) Then total length of ∂K_{w} is 4c_{w}. Let

p_{1}^{p_{2}}  ^{p_{3}}^{q_{23}}  ^{q_{32}}^{q_{21}}  ^{p}  ^{q_{31}}^{q_{12}}  ^{q_{13}} 2  4  3 5  6 1

Figure 1.6. Construction of diamond fractal

w ∈ Λ_{s}(c). Note that c_{w}/(1 − 2r) ≥ s ≥ c_{w}. Since {K_{w} ∩ K_{v}}_{v∈Λ_{s,w}(c)} provide a division of ∂K_{w}, it follows that #(Λ_{s}(c)) ≤ 4(1 + (1 − 2r)^{−1}). Therefore, for any r, 𝒮(c) is locally finite with respect to L.

The next example is the diamond fractal which has been introduced in [31]. This self-similar structure is not post critically finite but any self-similar scale is locally finite as in the post critically finite case.

Example 1.7.7 (Diamond fractal). Let p_{1}, p_{2}, p_{3} ∈ ℂ be vertices of a regular triangle with the length of edges 1, i.e. |p_{i} − p_{j}| = 1 if i ≠ j. Define p = (p_{1} + p_{2} + p_{3})/3. For i ∈ {1, 2, 3}, define F_{i}(z) = (z − p_{i})/3 + p_{i} and

F_{i+3}(z) = −\frac{1}{3} \frac{q_{i}}{\overline{q_{i}}} (\overline{z} − \overline{p_{i+3}}) + p_{i+3},

where q_{i} = p_{i}−p and p_{i+3} = (3p+p_{i})/4. Let q_{ij} = (2p_{i}+p_{j})/3 for any i, j ∈ {1, 2, 3}. If {i, j, k} = {1, 2, 3}, then F_{i+3} maps the regular triangle with vertices {p_{i}, p_{j}, p_{k}} to the regular triangle {p, p_{ij}, p_{ik}}.

There exists a unique nonempty compact set K satisfying K = ∪_{i=1}^{6}F_{i}(K). K is called the diamond fractal. The corresponding self-similar structure L =

(K, S, {F_{i}}_{i∈S}), where S = {1, …, 6}, is rationally ramified. In fact, the relation set R equals

\begin{matrix}\{(\{i, j\}, \{i, j\}, id, k, k + 3)|(i, j, k) = (1, 2, 3), (2, 3, 1), (3, 1, 2)\}∪ \\ \{(\{i\}, \{j\}, φ_{ij}, i + 3, j + 3)|(i, j) = (1, 2), (2, 3), (3, 1)\}, \end{matrix}

where id is the identity map and φ_{ij}(i) = j. By Theorem 1.6.1, any self-similar scale on Σ is locally finite with respect to L. Using Theorem 1.6.6, we see that, for a, b ∈ (0, 1)^{S}, 𝒮(a) ∼_{GE} 𝒮(b) if and only if there exists δ > 0 such that b_{i} = (a_{i})^{δ} for i = 1, 2, 3.

Figure 1.7. Diamond fractal

The rest of this section is devoted to a proof of Proposition 1.7.6.

Lemma 1.7.8. Let S be a finite set. Let f_{i} : ℝ → ℝ be an affine contraction for any i ∈ S, i.e. f_{i}(x) = r_{i}x + a_{i}, where |r_{i}| < 1. Let K be the self-similar set associated with {f_{i}}_{i∈S}. If f_{i}(K) ∩ f_{j}(K) = ∅ for any i, j ∈ S with i ≠ j, then ν_{1}(K) = 0, where ν_{1} is the 1-dimensional Hausdorff measure.

Proof. Define K_{ϵ} = {y|y ∈ ℝ, |x − y| ≤ ϵ for some x ∈ K}. Let y ∈ K_{ϵ}. Choose x ∈ K so that |x − y| ≤ ϵ. Then f_{i}(x) ∈ K and |f(y) − f(x)| ≤ |y − x| ≤ ϵ. Hence f_{i}(y) ∈ K_{ϵ}. This shows that f_{i}(K_{ϵ}) ⊆ K_{ϵ}. Let K_{ϵ}^{1} = ∪_{i∈S}f_{i}(K_{ϵ}). Then K_{ϵ}^{1} ⊆ K_{ϵ}. Since K_{ϵ} ≠ K, the uniqueness of the self-similar set implies K_{ϵ}^{1} ≠ K_{ϵ}. Therefore, if α = ν_{1}(K_{ϵ}^{1})/ν_{1}(K_{ϵ}), then α ∈ (0, 1). On the other hand, if we choose sufficiently small ϵ, then f_{i}(K_{ϵ}) ∩ f_{j}(K_{ϵ}) = ∅ for any i, j ∈ S with i ≠ j. Define K_{ϵ}^{m} inductively by K_{ϵ}^{m} = ∪_{i∈S}f_{i}(K_{ϵ}^{m−1}). Then ν_{1}(K_{ϵ}^{m}) = α^{m}ν_{1}(K_{ϵ}). Since K = ∩_{m≥0}K_{ϵ}^{m}, it follows that ν_{1}(K) = 0. □

Let L be a Sierpinski cross.

Proof of Proposition 1.7.6. If 1 − 2r = r^{m} for some m ∈ ℝ, then we can give the relation set R as in Example 1.7.5. Hence L is rationally ramified.

Next assume that L is rationally ramified with a relation set R. Let [R] = {Ω_{1}, …, Ω_{m}}, where Ω = (X_{i}, Y_{i}, φ_{i}, x(i), y(i)), be a relation set. Consider K_{8} ∩ K_{7} = F_{7}(L_{1}) ∩ F_{8}(L_{2}) = F_{8}(L_{2}), where L_{1} = [0, 1] and L_{2} = \{x + \sqrt{−1}|x ∈ [0, 1]\}. Define J = {i|x(i) ∈ σ_{7}(W_{∗}), y(i) ∈ σ_{8}(W_{∗}), ν_{1}(K_{x(i)}[X_{i}] ∩ K_{7} ∩ K_{8}) > 0}. By (1.5.1), K_{7} ∩ K_{8} ⊆ ∪_{i∈J}K_{x(i)}[X_{i}]. Choose i ∈ J so that K_{x(i)}[X_{i}] contains F_{7}(0) = F_{8}(\sqrt{−1}) and write X = X_{i}, Y = Y_{i}, φ = φ_{i}, x = x(i) and y = y(i) for simplicity. Since π^{−1}(F_{7}(0)) = {8(7)^{∞}, 7(1)^{∞}}, we see that x = 7(1)^{p}, X ⊆ W_{#}({1, 2, 3}), y = 8(7)^{q} and Y ⊆ W_{#}({7, 6, 5}). Note that ν_{1}(K[X]) > 0 and that K[X] is the selfsimilar set associated with {F_{w}}_{w∈X}. By Lemma 1.7.8, F_{w}(K[X]) ∩ F_{v}(K[X]) ≠ ∅ for some w, v ∈ X. Note that K[X] ⊆ L_{1} = [0, 1]. The intersection F_{w}(L_{1})∩F_{v}(L_{1}) contains a pair of points {π(w_{∗}(1)^{∞}), π(v_{∗}(3)^{∞}} if it is not empty. Hence there exists a w ∈ X such that w = (3)^{m}. This implies π((3)^{∞}) = 1 ∈ K[X]. Using the same arguments, we also obtain that π(5^{∞}) = 1 + \sqrt{−1} ∈ K[Y]. Since 1 + \sqrt{−1} and 1 are the most right points in K[X] and K[Y] respectively, F_{8(7)^{p}}(1 + \sqrt{−1}) = F_{7(1)^{q}}(1). Therefore, 1 − 2r = r^{q+1−p}. Since 0 < 1 − 2r < 1, it follows that q + 1 − p ≥ 1. □

CHAPTER 2 Construction of Distances

2.1. Distances associated with scales

We have studied the scale and the associated family of “balls” {U_{s}(x)}_{x∈X,s∈(0,1]} in the previous sections. Can this family of “balls”be thought of as real balls with respect to any distance? The next three sections are devoted to answer this question. In this section, we will define a pseudodistance on a self-similar set associated with a scale and consider when this pseudodistance is a distance.

As in the previous sections, S is a finite set, 𝒮 = {Λ_{s}}_{0<s≤1} is a right-continuous scale on Σ = Σ(S) whose gauge function is l and L = (K, S, {F_{i}}_{i∈S}) is a self-similar structure. Moreover, we assume that K is connected in the following sections.

Definition 2.1.1. A sequence of words, (w(1), …, w(m)),where w(i) ∈ W_{∗} for any i, is called a chain of L if and only if K_{w(i)} ∩ K_{w(i+1)} ≠ ∅ for i = 1, …, m − 1. We use CH to denote the collection of all chains of L. A chain (w(1), …, w(m)) is said to be a chain between x and y for x, y ∈ K if and only if x ∈ K_{w(1)} and y ∈ K_{w(m)}. The collection of all chains between x and y is denoted by CH(x, y).

Since K is assumed to be connected, CH(x, y) ≠ ∅ for any x, y ∈ K. See [28, Theorem 1.6.2].

Proposition 2.1.2. For x, y ∈ K, we define D_{𝒮}(x, y) by

D_{𝒮}(x, y) = inf{∑_{i=1}^{m} l(w(i))|(w(1), …, w(m)) ∈ CH(x, y)}.

Then D_{𝒮}(·, ·) is a pseudodistance on K: D_{𝒮}(x, y) = D_{𝒮}(y, x) ≥ 0 for any x, y, D_{𝒮}(x, x) = 0 for any x ∈ K and D_{𝒮}(x, z) ≤ D_{𝒮}(x, y)+D_{𝒮}(y, z) for any x, y, z ∈ K. Also D_{𝒮}(F_{w}(x), F_{w}(y)) ≤ l(w) for any x, y ∈ K. Moreover, if D_{𝒮}(·, ·) is a distance on K, then it is compatible with the original topology of K.

Proof. It is straight forward to see that D_{𝒮} is a pseudodistance on K by its definition. Since both F_{w}(x) and F_{w}(y) belong to K_{w}, D_{𝒮}(F_{w}(x), F_{w}(y)) ≤ l(w). Now assume that D_{𝒮} is a distance on K, i.e. D_{𝒮}(x, y) ≥ 0 for any x, y ∈ K. Note that {U_{s}(x)}_{0<s≤1} is a system of fundamental neighborhoods of x with respect to the original topology. Let d be a distance on K which gives the original topology of K. Suppose that d(x_{n}, x) → 0 as n → ∞ for a sequence {x_{n}}_{n≥1}. Then, for any s > 0, x_{n} belongs to U_{s}(x) for sufficiently large n. Hence D_{𝒮}(x, x_{n}) ≤ 2s for sufficiently large n. This implies that D_{𝒮}(x, x_{n}) → 0 as n → ∞. Conversely assume that D_{𝒮}(x_{n}, x) → 0 as n → ∞. Let y be an accumulating point of {x_{n}}_{n≥1} with respect to d: there exists a subsequence {y_{m}}_{m≥1} of {x_{n}}_{n≥1} such that d(y_{m}, y) → 0 as m → ∞. Since D_{𝒮}(y_{m}, x) → 0 as m → ∞, we see that D_{𝒮}(x, y) = 0. Hence x = y. Now the compactness of (K, d) implies that d(x_{n}, x) → 0 as n → ∞. □

Definition 2.1.3. D_{𝒮} is called the pseudodistance on K associated with the scale 𝒮. In particular, if 𝒮 = 𝒮(a) for a ∈ (0, 1)^{S}, then we write D_{𝒮} = D_{a}.

Remark. If 𝒮 = 𝒮(a) for a ∈ (0, 1)^{S}, then D_{𝒮} coincides with the standard pseudodistance on K with poly ratio a defined by Kameyama [26].

Notation. Let d be a (pseudo)distance on K. For x ∈ K and r > 0, we define B_{r}(x, d) = {y|y ∈ K, d(x, y) ≤ r). Also diam(A, d) = sup_{x,y∈A} d(x, y) for A ⊆ K.

B_{r}(x, d) is the r-ball around x with respect to d and diam(A, d) is the diameter of A with respect to d. A ball with respect to the pseudodistance D_{𝒮} always contains a “ball” associated with the scale 𝒮 as follows.

Proposition 2.1.4. For any n ∈ ℕ ∪ {0}, any s ∈ (0, 1] and any x ∈ K,

U_{s}^{(n)}(x) ⊆ B_{(n+1)s}(x, D_{𝒮}),

Proof. Let y ∈ U_{s}^{(n)}(x). Then there exists a chain (w(1), …, w(m)) between x and y such that m ≤ n + 1 and w(j) ∈ Λ_{s} for any j. Since l(w(j)) ≤ s for any j, it follows that ∑_{j=1}^{m} l(w(j)) ≤ (n + 1)s. □

In general, we have the next equivalence between conditions concerning a distance and a pseudodistance associated with a scale.

Proposition 2.1.5. Let d be a distance on K and let β > 0. Then the following four conditions are equivalent.

(1) K_{s}(x) ⊆ B_{βs}(x, d) for any x ∈ K and any s ∈ (0, 1].

(2) U_{s}^{(n)}(x) ⊆ B_{(n+1)βs}(x, d) for any n ≥ 0, any x ∈ K and any s ∈ (0, 1].

(3) d(x, y) ≤ βD_{𝒮}(x, y) for any x, y ∈ K.

(4) diam(K_{w}, d) ≤ βl(w) for any w ∈ W_{∗}.

In particular, if any of the four conditions above is satisfied, D_{𝒮} is a distance on K.

Recall that K_{s}(x) = U_{s}^{(0)}(x).

Proof. (1) ⇒ (3): Let (w(j))_{j=1,...,m} ∈ CH(x, y). Choose x_{j} ∈ K_{w(j)} ∩ K_{w(j+1)} for j = 1, …, m − 1. Set x_{0} = x and x_{m} = y. Then x_{j} ∈ U_{l(w(j))}(x_{j−1}) ⊆ B_{βl(w(j))}(x_{j−1}) for j = 1, …, m. Hence d(x_{j−1}, x_{j}) ≤ βl(w(j)). Summing these inequalities for j = 1 to j = m, we obtain d(x, y) ≤ βD_{𝒮}(x, y).

(3) ⇒ (2): Since B_{s}(x, D_{𝒮}) ⊆ B_{βs}(x, d), Proposition 2.1.4 suffices to see the claim.

(2) ⇒ (1): Obvious

(3) ⇒ (4): Let x and y belong to K_{w}. Since D_{𝒮}(x, y) ≤ l(w), it follows that d(x, y) ≤ βl(w).

(4) ⇒ (1): Let y ∈ U_{s}^{(0)}(x). Then x, y ∈ K_{w} for some w ∈ Λ_{s}. Since d(x, y) ≤ βl(w), we obtain (1). □

If we can find one elliptic scale 𝒮_{∗} where D_{𝒮_{∗}} is a distance, then for any elliptic scale 𝒮, D_{𝒮^{α}} is a distance for some α > 0. To give detailed version of such a result, we need the following definition.

Definition 2.1.6. Let 𝒮 be a scale on Σ and let l be its gauge function. For w ∈ W_{∗}, define l_{w} : W_{∗} → (0, 1] by l_{w}(v) = l(wv)/l(w). We denote the scale whose gauge function is l_{w} by 𝒮_{w}.

In the above definition, it is obvious that 𝒮_{w} is actually a (right-continuous) scale. In the followings, we use 𝒮_{w}^{α} to denote (𝒮^{α})_{w} for α > 0 and w ∈ W_{∗}. Note that (𝒮^{α})_{w} = (𝒮_{w})^{α}.

Proposition 2.1.7. Let 𝒮_{∗} be a scale on Σ with (EL2). Suppose that D_{𝒮_{∗}} is a distance on K. If a scale 𝒮 satisfies (EL1), then there exist α > 0 and β > 0 such that D_{𝒮_{w}^{α}} (x, y) ≥ βD_{𝒮_{∗}} (x, y) for any x, y ∈ K and any w ∈ W_{∗}. In particular, D_{𝒮_{w}^{α}} is a distance on K.

Proof. Let l_{∗} be the gauge function of 𝒮_{∗}. By (EL2), there exists γ ∈ (0, 1) and β > 0 such that l_{∗}(v) ≤ βγ^{|v|} for any v ∈ W_{∗}. Also if l is the gauge function of 𝒮, then (EL2) implies that there exists β_{1} ∈ (0, 1) such that l_{w}(v) ≥ (β_{1})^{|v|} for any v, w ∈ W_{∗}. Therefore if (β_{1})^{α} ≥ γ, then diam(K_{v}, D_{𝒮_{∗}}) ≤ β^{−1}l_{w}(v) for any v ∈ W_{∗}. By Proposition 2.1.5, we see that D_{𝒮_{w}^{α}} (x, y) ≥ βD_{𝒮_{∗}} (x, y). □

Next theorem gives a topological sufficient condition for D_{𝒮} being a distance. By virtue of this result, for any locally finite scale 𝒮 on a rationally ramified selfsimilar structures, D_{𝒮^{α}} is shown to be a distance for some α > 0 in the next section.

Theorem 2.1.8. Let 𝒮 = {Λ_{s}}_{0<s≤1} be a scale on Σ. Assume the existence of n ∈ ℕ satisfying the following two conditions (D1) and (D2): (D1) If w ∈ Λ_{s}, τ ∈ W_{n}, v ∈ Λ_{s,w} and K_{vτ} ∩ K_{w} ≠ ∅, then K_{vτ} ∩ K_{v′} = ∅ for any v′ ∈ Λ_{s}\Λ_{s,w}. (D2) Let l be the gauge function of 𝒮. Set β = (\sqrt{17} − 1)/4. Then l(wτ) ≥ βl(w) for any w ∈ W_{∗} and any τ ∈ W_{n}. Then for any x, y ∈ K,

inf{s|y ∈ U_{s}^{(3)}(x)} ≤ D_{𝒮}(x, y) ≤ 4 inf{s|y ∈ U_{s}^{(3)}(x)}.

In particular, D_{𝒮} is a distance on K. Moreover, for any s ∈ (0, 1] and any x ∈ K,

B_{s}(x, D_{𝒮}) ⊆ U_{s}^{(3)}(x) ⊆ B_{4s}(x, D_{𝒮}).

Note that 0 < β < 1.

The condition (D1) is shown to hold if 𝒮 is intersection type finite in the next section. See the next section for the notion of “intersection type finite”.

To prove the above theorem, we need several lemmas.

Lemma 2.1.9. For w ∈ Λ_{s}, we define U_{s}(w) = K(Λ_{s,w}) = K(W (Λ_{s}, K_{w})). Assume that (D1) is satisfied. If w ∈ Λ_{s}, τ ∈ W_{n}, v ∈ Λ_{s,w}, vτ ∈ Λ_{s′} and K_{vτ} ∩ K_{w} ≠ ∅, then U_{s′}(vτ) ⊆ U_{s}(w).

Proof. Let v′′ ∈ Λ_{s′,vτ}. Since K_{vτ} ∩ K_{v′} = ∅ for any v′ ∈ Λ_{s,w}, there exists w′ ∈ Λ_{s,w} such that w′ ≥ v′′. Therefore, K_{v′′} ⊆ K_{w′} ⊆ U_{s}(w). □

Lemma 2.1.10. Assume (D1) and (D2). Let (w, v) be a chain of L. If w ∈ Λ_{s}, v ∈ Λ_{s′} and βl(w) ≥ l(v), then U_{s′}(v) ⊆ U_{s}(w).

Proof. If |v| ≤ n, then l(v) ≥ βl(∅) ≥ βl(w) ≥ l(v). Therefore, 1 = l(∅) = l(w). Since w ∈ Λ_{s}, we see that w = ∅ and s = 1. Hence U_{s′}(v) ⊆ U_{s}(w) = K. Assume that |v| > n. Let v = v′z for z ∈ W_{n}. Then l(v) ≥ βl(v′). This implies l(w) ≥ l(v′). Therefore, v = v_{∗}ττ′ for v_{∗} ∈ Λ_{s}, τ ∈ W_{n} and τ′ ∈ W_{∗}. Since K_{v_{∗}x} ∩ K_{w} ≠ ∅, Lemma 2.1.9 implies that U_{l(v_{∗}τ)}(v_{∗}x) ⊆ U_{s}(w). Note that s′ ≤ l(v_{∗}τ). Hence U_{s′}(v) ⊆ U_{s}(w). □

Lemma 2.1.11. Assume (D1) and (D2). Let (v, w, τ) be a chain of L. If βl(w) < l(v) and βl(w) < l(τ), then there exists a chain (v′, τ′) such that v′ ≥ v, τ′ ≥ τ and l(v′) + l(τ′) < l(v) + l(w) + l(τ).

Proof. If |w| ≤ n, then let v′ = τ′ = ∅. By (D2), (1 + 2β)β ≥ 2 and l(w) ≥ β. Therefore, l(v) + l(w) + l(τ) > (1 + 2β)β ≥ 2 = l(v′) + l(τ′). Hence we may assume that |w| > n. Let w = w_{1} … w_{m}. Set w_{∗} = w_{1} … w_{m−n} and define s = l(w_{∗}). If l(v) ≥ s, then we may find v_{∗} ∈ W_{∗} such that v ≥ v_{∗}, K_{v_{∗}} ∩ K_{w} ≠ ∅ and v_{∗} ∈ Λ_{s}. If l(v) < s, then there exists a unique v_{∗} such that v_{∗} > v and v_{∗} ∈ Λ_{s}. Also we define τ_{∗} in the same way as v_{∗}. Since (1 + 2β)β ≥ 2 and (1 + β)β ≥ 1 by (D2),

\begin{aligned}l(w) + l(v) & > (1 + β)βl(w_{∗}) ≥ l(w_{∗}) ≥ l(v_{∗}) \\ l(w) + l(τ) & > (1 + β)βl(w_{∗}) ≥ l(w_{∗}) ≥ l(τ_{∗}) \\ l(v) + l(w) + l(τ) & > (1 + 2β)βl(w_{∗}) ≥ 2l(w_{∗}) ≥ l(v_{∗}) + l(τ_{∗}) \end{aligned} ; (2.1.1)

Since w_{∗} ∈ Λ_{s,v_{∗}}, K_{w} ∩K_{v_{∗}} ≠ ∅ and K_{w} ∩K_{τ_{∗}} ≠ ∅, (D1) implies that K_{v_{∗}} ∩K_{τ_{∗}} ≠ ∅. Define v′ = max{v_{∗}, v} and τ′ = max{τ_{∗}, τ}. Then by (2.1.1), (v′, τ′) satisfies the desired properties. □

Proof of Theorem 2.1.8. Define

\begin{aligned}CH^{s}(x, y) & = \{(w(j))_{j=1,...,m}|(w(j))_{j=1,...,m} ∈ CH(x, y), min_{j=1,...,m} l(w(j)) ≥ s\} \\ F (s) & = inf\{∑_{j=1}^{m} l(w(j))|(w(j))_{j=1,...,m} ∈ CH^{s}(x, y)\} \end{aligned}

for any s > 0 and any x, y ∈ K. Then F (s) is monotonically decreasing and D_{𝒮}(x, y) = lim_{s↓0} F (s). Note that we may only consider chains without loops in the definition of F (s). Hence there exists (w(j))_{j=1,...,m} ∈ CH^{s}(x, y) which attains the infimum. Set s_{j} = l(w(j)) and U_{j} = U_{s_{j}}(w(j)) for j = 1, …, m. If 1 < j < m, then Lemma 2.1.11 implies that βs_{j} ≥ s_{j−1} or βs_{j} ≥ s_{j+1}. Hence by Lemma 2.1.10, U_{j−1} ⊆ U_{j} or U_{j+1} ⊆ U_{j}. Therefore, there exists j_{∗} such that 1 ≤ j_{∗} ≤ m + 1 and

U_{1} ⊆ U_{2} ⊆ … ⊆ U_{j_{∗}−1}, U_{j_{∗}} ⊇ … ⊇ U_{m−1} ⊇ U_{m}.

Let s_{∗} = max{s_{j_{∗}−1}, s_{j_{∗}}}. Since K_{w(j_{∗}−1)} ∩ K_{w(j_{∗})} ≠ ∅, x ∈ U_{1} and y ∈ U_{m}, we see that y ∈ U_{s_{∗}}^{(3)}(x). Therefore, F (s) = ∑_{j=1}^{m} s_{j} ≥ s_{∗} ≥ inf{s|y ∈ U_{s}^{(3)}(x)}. Thus

D_{𝒮}(x, y) ≥ inf{s|y ∈ U_{s}^{(3)}(x)}. On the other hand, if y ∈ U_{s}^{(3)}(x), then there exists (w(1), w(2), w(3), w(4)) ∈ CH(x, y) such that w(j) ∈ Λ_{s} for j = 1, 2, 3, 4. Therefore, D_{𝒮}(x, y) ≤ 4s. Hence D_{𝒮}(x, y) ≤ 4 inf{s|y ∈ U_{s}^{(3)}(x)}.

Finally, since {U_{s}^{(3)}(x)}_{0<s≤1} is monotonically decreasing with respect to s and ∩_{0<s≤1}U_{s}^{(3)}(x) = {x}, we see that inf{s|y ∈ U_{s}^{(3)}(x)} > 0 if x ≠ y. □

2.2. Intersection type

Let L = (K, S, \{F_{i}\}_{i∈S}) be a self-similar structure satisfying K\\overline{V}_{0} ≠ ∅. A scale 𝒮 = {Λ_{s}}_{s∈(0,1]} is said to be intersection type finite if the topological types of K_{w} ∩ K_{v} for s ∈ (0, 1] and w, v ∈ Λ_{s} are finite. Under the assumption of a scale being intersection type finite, we can verify the conditions (D1) and (D2) in Theorem 2.1.8 and hence the associated pseudodistance is a distance for some power of the scale. See Theorem 2.2.6 for details.

First we define the notion of intersection pairs.

Definition 2.2.1. (1) Define IP(L) by

IP(L) = {(w, v)|w, v ∈ W_{#}, K_{w} ∩ K_{v} ≠ ∅, Σ_{w} ∩ Σ_{v} = ∅}.

(w, v) ∈ IP(L) is called an intersecting pair of L.

(2) Define

A = {(A, B, φ)|A and B are nonempty closed subsets of V_{0}

and φ : A → B is a homeomorphism between A and B}.

There exists a natural map from IP(L) → A.

Proposition 2.2.2. Define

Φ((w, v)) = ((F_{w})^{−1}(K_{w} ∩ K_{v}), (F_{v})^{−1}(K_{w} ∩ K_{v}), (F_{v})^{−1} ∘ F_{w}|_{(F_{w})^{−1}(K_{w}∩K_{v})})

for any (w, v) ∈ IP(L). Then Φ : IP(L) → A.

The image of an intersection pair under the map Φ is called the intersection type.

Definition 2.2.3. (1) Define IT (L) = Φ(IP(L)). An element of IT (L) is called an intersection type of L.

(2) Let 𝒮 = {Λ_{s}}_{0<s≤1} be a scale on Σ. Define

IP(L, 𝒮) = {(w, v)|(w, v) ∈ IP(L), w, v ∈ Λ_{s} for some s ∈ (0, 1]}

and

IT (L, 𝒮) = {Φ((w, v))|(w, v) ∈ IP(L, 𝒮)}

𝒮 is said to be intersection type finite with respect to L if and only if IT (L, 𝒮) is a finite set.

The following proposition is straight forward by definition.

Proposition 2.2.4. Let 𝒮 be a scale on Σ. If L is strongly finite and 𝒮 is intersection type finite with respect to L, then 𝒮 is locally finite.

The property of a scale being intersection type finite is preserved under the equivalence relation ∼_{GE}.

Proposition 2.2.5. Let 𝒮_{1} and 𝒮_{2} be elliptic scales. If 𝒮_{1} is intersection type finite and 𝒮_{1} ∼_{GE} 𝒮_{2}, then 𝒮_{2} is intersection type finite.

Proof. Set 𝒮_{1} = {Λ_{s}}_{0<s≤1} and 𝒮_{2} = {Λ′_{s}}_{0<s≤1}. Also let l be the gauge function of 𝒮_{1}. Suppose that w, v ∈ Λ′_{s}, that K_{w} ∩ K_{v} ≠ ∅ and that l(v) ≤ l(w). Since 𝒮_{1} ∼_{GE} 𝒮_{2} and 𝒮_{1} is elliptic, there exist n (which is independent of s, w and

v) such that v = v_{∗}τ and v_{∗} ∈ Λ_{l(w)} for some v_{∗}, τ ∈ W_{∗} with |τ| ≤ n. Therefore, Φ((w, v)) ∈ {Φ((w, v_{∗}z))||z| ≤ n}. Note that {Φ((w, v_{∗}z))||z| ≤ n} only depends on Φ((w, v_{∗})). Therefore, if 𝒮_{1} is intersection type finite, then so is 𝒮_{2}. □

Now we present the first main theorem of this section.

Theorem 2.2.6. Let 𝒮 be a scale on Σ with (EL1). If 𝒮 is intersection type finite, then there exists α > 0 such that D_{𝒮^{α}} is a distance on K and B_{s}(x, D_{𝒮^{α}}) ⊆ U_{s^{1/α}}^{(3)}(x) ⊆ B_{4s}(x, D_{𝒮^{α}}) for any s ∈ (0, 1] and any x ∈ K.

Proof. Let 𝒮 = {Λ_{s}}_{0<s≤1} and let l be its gauge function. First we show (D1). Since 𝒮 is intersection type finite, there exists compact subsets B_{1}, …, B_{m} ⊂ K such that Φ((w, v)) = (B_{i}, B_{j}, ϕ_{ij}) for any s ∈ (0, 1], any w ∈ Λ_{s} and any v ∈ Λ_{s,w}. Define W_{k,j} = {τ|τ ∈ W_{k}, K_{τ} ∩ B_{j} ≠ ∅} and K_{k,j} = ∪_{τ∈W_{k,j}}K_{τ} for any j. Since ∩_{k≥1}K_{k,j} = B_{j}, there exists n such that K_{n,j} ∩ B_{p} = ∅ for any j, p ∈ {1, …, m} with B_{j} ∩ B_{p} = ∅. This implies (D1).

Now note that 𝒮^{α} satisfies (D1) with the same n as 𝒮 for any α > 0. Since 𝒮 satisfies (EL1), there exists γ ∈ (0, 1) such that l(wv) ≥ γl(w) for any w ∈ W_{∗} and any v ∈ W_{n}. Choosing α so that γ^{α} ≥ β = (\sqrt{17} − 1)/4, we see that 𝒮^{α} satisfies (D2). Thus by Theorem 2.1.8, D_{𝒮^{α}} is a distance on K. □

The second main theorem of this section tells us that one can identify “intersection type” finite with “locally” finite in the rationally ramified case.

Theorem 2.2.7. Let L be a rationally ramified self-similar structure. Then an elliptic scale 𝒮 on Σ is intersection type finite if and only if 𝒮 is locally finite with respect to L.

Proof. Since L is strongly finite, if 𝒮 is intersection type finite, then 𝒮 is locally finite by Proposition 2.2.4. Conversely assume that 𝒮 is locally finite. Let R be the relation set of L. We may assume that R = [R] without loss of generality. Set 𝒮 = {Λ_{s}}_{0<s≤1}.

Let R′ be a subset of R_{L}. For (w, v) ∈ IP(L), define R(w, v, R′) =

{(Ω, (z, x_{0}, …, x_{m}), (z, y_{0}, …, y_{n}))|Ω = (X, Y, φ, x, y) ∈ R′,

(z, x_{0}, …, x_{m}) ∈ A_{X,x}(w), (z, y_{0}, …, y_{n}) ∈ A_{Y,y}(v),

y_{j} = φ(x_{j})for j = 1, …, min{m, n}}

Let η = (Ω, (z, x_{0}, …, x_{m}), (z, y_{0}, …, y_{n})) ∈ R(w, v, R′) with Ω = (X, Y, φ, x, y). Note that z = w_{1} … w_{N}, where N = inf{i|w_{i} ≠ v_{i}} − 1 and that the first symbols of x and y are w_{N+1} and v_{N+1} respectively. Define K(η, w), K(η, v) and ψ_{η} : K(η, w) → K(η, v) as follows. If m ≥ n, then we set K(η, w) = K_{x_{m}^{2}}[X], K(η, v) = K_{y_{n}^{2}y_{n+1}...y_{m}}[Y] and ψ_{η} = F_{y_{n}^{2}y_{n+1}...y_{m}} ∘ φ̃ ∘ (F_{x_{m}^{2}})^{−1}, where x_{m}^{2} and y_{n}^{2} are given in Lemma 1.5.16 and y_{j} = φ(y_{j}) for j = n + 1, …, m. If m < n, then K(η, w) = K_{x_{m}^{2}x_{m+1}...x_{n}}[X], K(η, v) = K_{y_{n}^{2}}[Y] and ψ_{η} = F_{y_{n}^{2}} ∘φ̃∘(F_{x_{m}^{2}x_{m+1}...x_{n}})^{−1}, where x_{j} = φ^{−1}(y_{j}) for j = m + 1, …, n. Note that F_{w}(K(η, w)) = K_{v}(K(η, v)) ⊆ K_{w} ∩ K_{v} and that ψ_{η} = F_{v}^{−1}∘F_{w}|_{K(η,w)} by Lemma 1.5.16.

Next, we define

IP(L, 𝒮, R′) = {(w, v)|(w, v) ∈ IP(L, 𝒮), R(w, v, R′) ≠ ∅}

and

IT (L, 𝒮, R′) = {(K(η, w), K(η, v), ψ_{η})|(w, v) ∈ IP(L, 𝒮, R′), η ∈ R(w, v, R′)},

where R′ is a subset of R_{L}. The first step of the proof is to show that IT (L, 𝒮, R) = IT (L, 𝒮, R_{1}) ∪ IT (L, 𝒮, R_{2}) is a finite set, where R_{1} and R_{2} are the same as in Theorem 1.6.7. First we consider IT (L, 𝒮, R_{2}). Let (w, v) ∈ IP(L, 𝒮) and let η = (Ω, (z, x_{0}, …, x_{m}), (z, y_{0}, …, y_{n})) ∈ R(w, v, R_{2}) with Ω = (X, Y, φ, x, y). Since 𝒮 is locally finite with respect to L, Theorem 1.6.1 implies that |y_{n+1} … y_{m}| or |x_{m+1} … x_{n}| (depending on m ≥ n or m < n) is uniformly bounded with respect to w, v and η. Also by Lemma 1.6.3, #(A_{X,x}(w)) is uniformly bounded with respect to Ω and w. Therefore, IT (L, 𝒮, R_{2}) is finite.

Secondly, let η = (Ω, (z, x_{0}, …, x_{m}), (z, y_{0}, …, y_{n})) ∈ R(w, v, R_{1}) with Ω = ({x_{∗}}, {y_{∗}}, φ, x, y). Then K(η, w) ∈ {π(σ^{i}(x(x_{∗})^{∞})|i = 1, 2, …} and K(η, v) ∈ {π(σ^{i}(y(y_{∗})^{∞})|i = 1, 2, …}. Since R_{1} is finite, IT (L, 𝒮, R_{1}) is finite. Thus it follows that IT (L, 𝒮, R) is a finite set.

To proceed to the next step, we need to define δ_{m}∘ … ∘δ_{1} for δ_{1}, …, δ_{m} ∈ A. For δ_{1} = (A_{1}, B_{1}, φ_{1}) and δ_{2} = (A_{2}, B_{2}, φ_{2}) ∈ A, define δ_{2}∘δ_{1} ∈ A by

δ_{2}∘δ_{2} = ((φ_{1})^{−1}(A_{2} ∩ B_{1}), φ_{2}(A_{2} ∩ B_{1}), φ_{2}∘φ_{1}|_{(φ_{1})^{−1}(A_{2}∩B_{1})}).

Then δ_{m}∘ … ∘δ_{1} is defined inductively by δ_{m}∘(δ_{m−1}∘ … ∘δ_{1}).

Now, let (w, v) ∈ IP(L, 𝒮) and let p ∈ K_{w} ∩ K_{v}. Choose s so that w, v ∈ Λ_{s}. Choose ω ∈ Σ_{w} ∩ π^{−1}(p) and τ ∈ Σ_{v} ∩ π^{−1}(p). By Proposition 1.5.13-(2), there exist Ω_{1}, …, Ω_{m} ∈ R and ω^{(1)}, …, ω^{(m+1)} ∈ Σ(S) which satisfies (AS1), (AS2) and (AS3). For some n, ω^{(n)} ∉ Σ_{v} but ω^{(n+1)} ∈ Σ_{v}. Recall the remark after Proposition 1.5.13. Set m_{j} = s(ω^{(j)}, τ). Then m_{j} < |v| for j = 1, …, n. Hence letting w(j) = ω_{1}^{(j)} … ω_{m_{j}}^{(j)}(= v_{1} … v_{m_{j}}), then l(w(j)) ≥ l(v_{1} … v_{|v|−1}) > s, where l is the gauge function of 𝒮. We may choose k_{j} > m_{j} so that v(j) = ω_{1}^{(j)} … ω_{k_{j}}^{(j)} ∈ Λ_{s} for any j = 1, …, n. (We set v(1) = w and v(n+1) = v.) Let Ω_{j} = (X_{j}, Y_{j}, φ_{j}, x(j), y(j)). Then, ω^{(j)} = w(j)x(j)x_{1}x_{2} … for some x_{1}x_{2} … ∈ Σ[X_{j}] and ω^{(j+1)} = w(j)y(j)y_{1}y_{2} …, where y_{i} = φ_{j}(x_{i}). Hence, for some M_{j} and N_{j}, η_{j} = (Ω_{j}, (w(j), x(j), x_{1}, …, x_{M_{j}}), (w(j), y(j), y_{1}, …, y_{M_{j}}) ∈ R(v(j), v(j + 1), R). Define ρ_{j} = (K(η_{j}, v(j)), K(η_{j}, v(j + 1)), ψ_{η_{j}}). Then ρ_{j} ∈ IT (L, 𝒮, R). Now, ρ_{n}∘ … ∘ ρ_{1} gives a fraction of Φ((w, v)) around F_{w}^{−1}(p). Therefore, Φ((w, v)) is a combination of elements in {δ_{1} ∘…∘δ_{n}|n ≤ max_{p∈K} #(π^{−1}(p)), δ_{i} ∈ IT (L, 𝒮, R))}. Since this set is finite, L is intersection type finite. □

Combining the last two theorems, we obtain the following fact.

Corollary 2.2.8. Let L be a rationally ramified self-similar structure. If an elliptic scale 𝒮 on Σ is locally finite, then D_{𝒮^{α}} is a distance on K for some α > 0. Moreover, B_{s}(x, D_{𝒮^{α}}) ⊆ U_{s^{1/α}}^{(3)}(x) ⊆ B_{4s}(x, D_{𝒮^{α}}) for any s ∈ (0, 1] and any x ∈ K. In particular, if L is post critically finite, then, for any elliptic scale on K, D_{𝒮^{α}} is a distance on K for some α > 0.

Proof. The first half is verified by using Theorems 2.2.6 and 2.2.7. About post critically finite self-similar structure, recall that any scale on K is locally finite. This suffices the conclusion. □

Remark. In [26], Kameyama has shown that there exists a self-similar scale α ∈ (0, 1)^{S} such that D_{α} is a distance on K for any critically finite self-similar set, which corresponds to post critically finite self-similar structure in our language. (His definition of self-similar sets allows that the contraction mappings are not injective.) The above corollary partially extends his result to rationally ramified case.

In the rest of this section, we will give several accounts about intersection pairs. Those results are rather technical but play important roles later.

Definition 2.2.9. Let Γ_{i} ⊆ W_{∗} for i = 1, 2. A bijection ψ : Γ_{1} → Γ_{2} is called an L-isomorphism between Γ_{1} and Γ_{2} if the following condition is satisfied: For w, v ∈ Γ_{1}, (w, v) ∈ IP(L) if and only if (ψ(w), ψ(v)) ∈ IP(L). If (w, v) ∈ IP(L) for w, v ∈ Γ_{1}, then Φ((w, v)) = Φ((ψ(w), ψ(v))).

Γ_{1} and Γ_{2} are said to be L-similar if there exists an L-isomorphism between Γ_{1} and Γ_{2}.

Proposition 2.2.10. Let Γ_{i} ⊆ W_{∗} for i = 1, 2 and let ψ be an L-isomorphism between Γ_{1} and Γ_{2}. Then there exists a homeomorphism ϕ between K(Γ_{1}) and K(Γ_{2}) such that ϕ|_{K_{w}} = F_{ψ(w)} ∘(F_{w})^{−1} for any w ∈ Γ_{1}. ϕ is called the L-similitude between K(Γ_{1}) and K(Γ_{2}) associated with ψ.

For Γ_{1}, Γ_{2} ⊆ W_{∗}, we say that ϕ : K(Γ_{1}) → K(Γ_{2}) is an L-similitude between K(Γ_{1}) and K(Γ_{2}) if and only if there exists a L-isomorphism ψ between Γ_{1} and Γ_{2} and ϕ is associated with ψ.

Proof. Let (w, v) ∈ IP(L) for w, v ∈ Γ_{1}. Since Φ((w, v)) = Φ((ψ(w), ψ(v))), it follows that F_{ψ(w)}∘(F_{w})^{−1} coincides with F_{ψ(v)}∘(F_{v})^{−1} on K_{w} ∩ K_{v}. Hence if ϕ = F_{ψ(w)}∘(F_{w})^{−1} on K_{w}, then ϕ is a well-defined homeomorphism between K(Γ_{1}) and K(Γ_{2}). □

Definition 2.2.11. Let n ∈ {0} ∪ ℕ. For (s_{1}, x_{1}), (s_{2}, x_{2}) ∈ (0, 1] × K, we write (s_{1}, x_{1}) ∼_{n} (s_{2}, x_{2}) if and only if there exists an L-isomorphism ψ between

Λ_{s_{1},x_{1}}^{n} and Λ_{s_{2},x_{2}}^{n} such that ψ(Λ_{s_{1},x_{1}}^{k}) = Λ_{s_{2},x_{2}}^{k} for any k = 0, 1, …, n. We call ψ the n-isomorphism between (s_{1}, x_{1}) and (s_{2}, x_{2}).

Note that (s_{1}, x_{1}) ∼_{n} (s_{2}, x_{2}) implies (s_{1}, x_{1}) ∼_{k} (s_{2}, x_{2}) for any 0 ≤ k ≤ n. It is easy to see that ∼_{n} is an equivalence relation.

Proposition 2.2.12. The relation ∼_{n} is an equivalence relation on (0, 1] × K for any n ≥ 0.

We can relate the notion of being intersection type finite with the number of equivalence classes under ∼_{n}.

Theorem 2.2.13. Let L be strongly finite. Then the following three conditions are equivalent.

(1) 𝒮 is intersection type finite with respect to L.

(2) ((0, 1] × K)/∼_{n} is a finite set for any n ∈ {0} ∪ ℕ.

(3) ((0, 1] × K)/∼_{n} is a finite set for some n ∈ {0} ∪ ℕ.

The following fact, which is used to show the above theorem, is straight forward.

Lemma 2.2.14. For (s, x) ∈ (0, 1] × K and n ∈ {0} ∪ ℕ, define J_{s,x}^{n} : Λ_{s,x}^{n} → {0, 1, …, n} by J_{s,x}^{n}(w) = min{k|w ∈ Λ_{s,x}^{k}} and define H_{s,x}^{n} : Λ_{s,x}^{n} × Λ_{s,x}^{n} → IT (L, 𝒮) ∪ {0, 1} by

H_{s,x}^{n}(w, v) = \begin{cases}Φ((w, v)) & if (w, v) ∈ IP(L), \\ 0 & if K_{w} ∩ K_{v} = ∅, \\ 1 & if w = v. \end{cases}

Then ψ is an n-isomorphism between (s_{1}, x_{1}) and (s_{2}, x_{2}) if and only if J_{s_{1},x_{1}}(w) = J_{s_{2},x_{2}}(ψ(w)) and H_{s_{1},x_{1}}(w, v) = H_{s_{2},x_{2}}(ψ(w), ψ(v)) for any w, v ∈ Λ_{s_{1},x_{1}}^{n}.

Proof of Theorem 2.2.13. (1) ⇒ (2): Assume that 𝒮 is intersection type finite with respect to L. Then by Proposition 2.2.4, 𝒮 is locally finite with respect to L. Hence Lemma 1.3.6 implies that #(Λ_{s,x}^{n}) is uniformly bounded with respect to (s, x) ∈ (0, 1] × K. Since IT (L, 𝒮) is a finite set, we only have finite number of choices of J_{s,x}^{n} and H_{s,x}^{n} up to n-isomorphisms. Therefore by Lemma 2.2.14, ((0, 1] × K)/∼_{n} is a finite set for any n ∈ {0} ∪ ℕ.

(2) ⇒ (3): This is obvious.

(3) ⇒ (1): We see that ((0, 1] × K)/∼_{0} is a finite set under (3). Since L is strongly

finite, #(Λ_{s,x}) is uniformly bounded with respect to (s, x) ∈ (0, 1]×K. Therefore if X = ∪_{(s,x)∈(0,1]×K}Im H_{s,x}^{0}, then Lemma 2.2.14 implies that X is a finite set. Note that IT (L, 𝒮) ⊆ X. Thus we have #(IT (L, 𝒮)) is finite. □

2.3. Qdistances adapted to scales

As is seen in the last section, D_{𝒮} is not always a distance even if a scale 𝒮 is elliptic and locally finite. Instead we sometimes managed to show that D_{𝒮^{α}} is a distance for some α > 0. In such a case, if d(x, y) = (D_{𝒮^{α}} (x, y))^{1/α}, then d has the same scaling ratio as the scale 𝒮 but d is not a distance. Considering such a situation, we will introduce the notion of an α-qdistandce in this section.

Definition 2.3.1. Let X be a set. For α > 0, d : X × X → [0, ∞) is called αqdistance on X if and only if d(x, y)^{α} is a distance on X. Also d is called a qdistance on X if it is an α-qdistance on X for some α > 0. We say d_{1} and d_{2} are equivalent if there exist c_{1} > 0 and c_{2} > 0 such that c_{1}d_{1}(x, y) ≤ d_{2}(x, y) ≤ c_{2}d_{1}(x, y) for any x, y ∈ X.

Remark. We may give more general definition of a qdistance: let f : [0, ∞) → [0, ∞) satisfy that f(x) < f(y) if x, y ∈ [0, ∞) and x < y, lim_{x↓0} f(x) = f(a) for any a ∈ [0, ∞) and that f(0) = 0. Then d : X × X → [0, ∞) is called f-qdistance on X if and only if f(d(x, y)) is a distance on X. In this paper, however, we do not need such an generality. So we restrict ourselves to the case where f(x) = x^{α} for some α > 0.

The symbol “q” of qdistance represents the prefix “quasi”. We do not use the word “quasidistance” to avoid confusion with the existent notion of quasidistance (or quasimetric) which has been defined as follows: d : X × X → [0, +∞) is called a C-quasidistance (or quasimetric) for C > 0 if and only if d(x, y) = 0 is equivalent to x = y, d(x, y) = d(y, x) for any x, y and

d(x, z) ≤ C(d(x, y) + d(y, z))

for any x, y, z. A qdistance is is a quasidistance. (In fact, an α-qdistance is a 2^{1/α−1}- quasidistance.) The immediate converse itself is not true. We have, however, the following modification of the converse.

Proposition 2.3.2. Let d be a quasidistance on a set X. Then d is equivalent to an α-qdistance D for some α > 0, i.e. there exist positive constants c_{1} and c_{2} such that c_{1}d(x, y) ≤ D(x, y) ≤ c_{2}d(x, y) for any x, y ∈ X.

Proof. This proposition is a version of [23, Proposition 14.5] in terms of “qdistance”. □

If d is a qdistance, lim_{n→∞} d(x_{n}, x) = 0 implies lim_{n→∞} d(x_{n}, y) = d(x, y) for any y. This is not the case in general for a quasidistance.

If d is an α-qdistance, then d is an α′-qdistance for any α′ ∈ (0, α], because a^{s} ≤ b^{s} +c^{s} for any a, b, c, ∈ ℝ with a ≤ b+c and any s ∈ (0, 1]. In particular, if d is an α-qdistance for α > 1, then d is a distance. Thus, we will consider α-qdistances for α ∈ (0, 1].

For an α-qdistance d on a set X, we always associate the topology given by the distance d^{α}. Also we may define Hausdorff measures and Hausdorff dimensions of subsets of X in the same manner as in the case of distance as follows.

Definition 2.3.3. Let d be a qdistance on X. Then for any A ⊆ X, we define

H_{δ}^{s}(A) = inf{∑_{i≥1} diam(E_{i})^{s}|A ⊆ ∪_{i≥1}E_{i}, diam(E_{i}) ≤ δ}

for any δ > 0 and s ≥ 0, where diam(E) = sup_{x,y∈A} d(x, y). Also we define H^{s}(A) = lim_{δ↓0} H_{δ}^{s}(A). H^{s} is called the s-dimensional Hausdorff measure with respect to the qdistance d. Also let

dim_{H}(A, d) = sup{s|H^{s}(A) = ∞} = inf{s|H^{s}(A) = 0}

for any A ⊆ X. dim_{H}(A, d) is called the Hausdorff dimension of A with respect to the qdistance d.

As in the case of distances, H^{s} is a complete Borel regular measure on X for any s ≥ 0.

Hereafter in this section, S is a non-empty finite set and L = (K, S, {F_{i}}_{i∈S}) is a self-similar structure with K\\overline{V}_{0} ≠ ∅. Also 𝒮 is a right-continuous scale.

Definition 2.3.4. A qdistance d on K is said to be adapted to a scale 𝒮 if and only if there exists β_{1}, β_{2} > 0 and n ∈ ℕ such that

B_{β_{1}s}(x, d) ⊆ U_{s}^{(n)}(x) ⊆ B_{β_{2}s}(x, d)

for any x ∈ K and any s ∈ (0, 1].

For example, the distance D_{𝒮} given in Theorem 2.1.8 is adapted to the scale 𝒮 with n = 3, β_{1} = 1 and β_{2} = 4.

Proposition 2.3.5. If d is a qdistance on K which is adapted to a scale 𝒮, then the topology on K given by d is the same as the original topology of K.

Proof. Note that {U_{s}^{(n)}(x)}_{0<s≤1} is a fundamental system of neighborhoods of x for any x ∈ K. This immediately imply the desired statement. □

Hereafter, we always assume that the topology of K given by a qdistance d is the same as the original topology of K.

First we give an extension of Moran-Hutchinson’s theorem on the Hausdorff dimension of self-similar sets.

Theorem 2.3.6. Let 𝒮 be a scale on Σ which satisfies (EL1) and let l be the gauge function of 𝒮. Assume that 𝒮 is locally finite and that there exist positive constants c_{1}, c_{2}, γ and a Borel regular measure ν on K such that c_{1}l(w)^{γ} ≤ ν(K_{w}) ≤ c_{2}l(w)^{γ} for any w ∈ W_{∗}. Also assume that d is a qdistance on K which is adapted to 𝒮. Then, there exist positive constants c_{3} and c_{4} such that

(2.3.1)  c_{3}H^{γ}(A) ≤ ν(A) ≤ c_{4}H^{γ}(A)

for any Borel set A ⊆ K and

(2.3.2)  c_{3}r^{γ} ≤ ν(B_{r}(x, d)) ≤ c_{4}r^{γ}

for any x ∈ K and any r > 0. In particular, dim_{H}(K, d) = γ.

Proof. First we show that ν is elliptic. Since 𝒮 satisfies (EL1), there exists β_{1} > 0 such that l(wi) ≥ β_{1}l(w) for any w ∈ W_{∗} and any i ∈ S. Therefore ν(K_{wi}) ≥ c_{1}l(wi)^{γ} ≥ c_{1}(β_{1}l(w))^{γ}. Hence ν is elliptic. By Theorem 1.2.4, we see that ν ∈ M(K). This implies

ν(U_{s}^{(n)}(x)) = ∑_{w∈Λ_{s,x}^{n}} ν(K_{w}).

for any s ∈ (0, 1] and any x ∈ K. Since μ is elliptic and 𝒮 is locally finite, there exists positive constants c_{5} and c_{6} such that

c_{5}s^{γ} ≤ ν(U_{s}^{(n)}(x)) ≤ c_{6}s^{γ}

for any s ∈ (0, 1] and any x ∈ K. As d is adapted to 𝒮, this immediately shows (2.3.2). By (2.3.2), using the mass distribution principle (i.e. Frostman’s lemma, see [28, Lemma 1.5.5]), we conclude that there exists c_{4} > 0 such that ν(A) ≤ c_{4}H^{γ}(A) for any Borel set A. Next fix w ∈ W_{∗}. For sufficiently small s, define Z_{w} = {v|v ∈ Λ_{s}, v ≤ w}. Then K_{w} = ∪_{v∈Z_{w}}K_{v}. Note that there exist c′ > 0 and c′′ > 0 such that diam(K_{v}, d)^{γ} ≤ c′s^{γ} ≤ c′′ν(K_{v}) for any s ∈ (0, 1] and v ∈ Λ_{s}. Therefore, ∑_{v∈Z_{w}} diam(K_{v}, d)^{γ} ≤ c′′ ∑_{v∈Z_{w}} ν(K_{v}) = c′′ν(K_{w}) because ν ∈ M(K). Since max_{v∈Z_{w}} diam((K_{v}, d)) → 0 as s → 0, it follows that H^{γ}(K_{w}) ≤ c′′ν(K_{w}). By [28, Theorem 1.4.10], we obtain (2.3.1). □

In general, it is difficult to find a measure ν satisfying the assumption of the above theorem. However, if 𝒮 is a scale induced by an elliptic measure μ, then we may let ν = μ and have γ = 1. Also there is an obvious choice of ν and γ in the case of a self-similar scale. The following corollary corresponds to the classical Moran-Hutchinson theorem on the Hausdorff dimension of a self-similar set with the open set condition. See [28, Section 1.5]. Also see [35, 24].

Corollary 2.3.7. Let a = (a_{i})_{i∈S} ∈ (0, 1)^{S}. Assume that 𝒮(a) is locally finite and that d is a qdistance on K which is adapted to 𝒮(a). Then the results of Theorem 2.3.6 holds, where γ is the unique constant which satisfies ∑_{i∈S}(a_{i})^{γ} = 1 and ν is the self-similar measure with weight ((a_{i})^{γ})_{i∈S}.

Definition 2.3.8. (1) Let 𝒮 be a scale on Σ. For n ≥ 1, define

δ_{𝒮}^{(n)}(x, y) = inf{s|y ∈ U_{s}^{(n)}(x)}

for any x, y ∈ K.

(2) Let d be a qdistance. We say that d is n-adapted to 𝒮 if and only if there exist c_{1}, c_{2} > 0 such that c_{1}d(x, y) ≤ δ_{𝒮}^{(n)}(x, y) ≤ c_{2}d(x, y) for any x, y ∈ K.

Obviously, a qdistance d is adapted to 𝒮 if and only if it is n-adapted to 𝒮 for some n ≥ 1. If no confusion may occur, we omit 𝒮 in δ_{𝒮}^{(n)} and write δ^{(n)}. The following proposition is immediate from the definition.

Proposition 2.3.9. Let 𝒮 be a scale on Σ. For any n ≥ 1 and any x, y ∈ K, δ^{(n)}(x, y) = δ^{(n)}(y, x), δ^{(n)}(x, y) ≥ 0 and the equality holds if and only if x = y.

Lemma 2.3.10. Let 𝒮 be a scale on Σ. Fix n ∈ ℕ and α > 0. Then the following three conditions are equivalent: (A) There exists an α-qdistance which is n-adapted to 𝒮. (B) D_{𝒮^{α}} is a distance and (D_{𝒮^{α}})^{1/α} is n-adapted to 𝒮. (C) D_{𝒮^{α}} is a distance and (D_{𝒮^{α}})^{1/α} is m-adapted to 𝒮 for any m ≥ n.

Moreover, let d be an α-qdistance. Then d is n-adapted to 𝒮 if and only if (D_{𝒮^{α}})^{1/α} is an α-qdistance which is n-adapted to 𝒮 and d is equivalent to (D_{𝒮^{α}})^{1/α}.

Proof. (A) ⇒ (B) Let d be an α-qdistance which is n-adapted to 𝒮. Then d^{α} is a distance and there exist c_{1}, c_{2} > 0 such that

B_{c_{1}s}(x, d^{α}) ⊆ U_{s^{1/α}}^{(n)}(x) ⊆ B_{c_{2}s}(x, d^{α})

for any x and any s. Applying Proposition 2.1.5, we obtain d(x, y)^{α} ≤ βD_{𝒮^{α}} (x, y) for any x, y, where β = c_{2}/(n+1). In particular, D_{𝒮^{α}} is a distance and B_{s}(x, D_{𝒮^{α}}) ⊆ B_{βs}(x, d^{α}). Moreover, by Proposition 2.1.4, U_{s^{1/α}}^{(n)}(x) ⊆ B_{(n+1)s}(x, D_{𝒮^{α}}). Hence (D_{𝒮^{α}})^{1/α} is n-adapted to 𝒮. (B) ⇒ (C) By Proposition 2.1.4, U_{s^{1/α}}^{(m)}(x) ⊆ B_{(m+1)s}(x, D_{𝒮^{α}}). This along with the fact that U_{s}^{(n)}(x) ⊆ U^{(m+1)} shows that (D_{𝒮^{α}})^{1/α} is m-adapted. (C) ⇒ (A) This is obvious.

The remaining statement is easily verified from the arguments in “(A) ⇒ (B)”. □

Theorem 2.3.11. Let 𝒮 be a scale on Σ and let n ∈ ℕ. The following six properties are equivalent: (A) δ^{(n)} is a quasidistance. (B) There exists a qdistance which is n-adapted to 𝒮. (C) There exists α > 0 such that D_{𝒮^{α}} is a distance and (D_{𝒮^{α}})^{1/α} is n-adapted to 𝒮. (D) There exists α > 0 such that D_{𝒮^{α}} is a distance and (D_{𝒮^{α}})^{1/α} is m-adapted to 𝒮 for any m ≥ n. (E) For any m ≥ n, there exists c > 0 such that cδ^{(m)}(x, y) ≥ δ^{(n)}(x, y) for any x, y ∈ K. (F) There exists c > 0 such that cδ^{(2n+1)}(x, y) ≥ δ^{(n)}(x, y) for any x, y ∈ K.

Proof. (A) ⇒ (B) Proposition 2.3.2 suffices this implication. (B) ⇒ (C) ⇒ (D) This is immediate by Lemma 2.3.10. (D) ⇒ (E) Let d = (D_{𝒮^{α}})^{1/α}. Since d is both m and n- adapted to 𝒮, there exist β_{1}, β_{2} > 0 such that β_{1}δ^{(m)}(x, y) ≥ β_{2}d(x, y) ≥ δ^{(n)}(x, y) for any x, y. (E) ⇒ (F) This is obvious. (F) ⇒ (A) Let x, y and z belong to K. If t > max δ^{(n)}(x, y), δ^{(n)}(y, z), then y ∈ U_{t}^{(n)}(x) and z ∈ U_{t}^{(n)}(y). Hence x ∈ U_{t}^{(2n+1)}(z). This shows that δ^{(2n+1)}(x, z) ≤ δ^{(n)}(x, y) + δ^{(n)}(y, z). By (5), δ^{(n)}(x, z) ≤ c(δ^{(n)}(x, y) + δ^{(n)}(y, z)). □

By the above theorem, a qdistance which is adapted to a scale 𝒮 is essentially (D_{𝒮^{α}})^{1/α}. Also, qdistances which are adapted to a scale 𝒮 are all equivalent.

Corollary 2.3.12. Let 𝒮 be a scale on Σ.

(1) There exists a qdistance which is adapted to 𝒮 if and only if (D_{𝒮^{α}})^{1/α} is a α-qdistance which is adapted to 𝒮 for some α > 0.

(2) Let d be a qdistance. Then d is adapted to 𝒮 if and only if (D_{𝒮^{α}})^{1/α} is a α-qdistance which is adapted to 𝒮 for some α > 0 and d is equivalent to (D_{𝒮^{α}})^{1/α}.

(3) Let d_{1} be a qdistances adapted to 𝒮. Then a qdistance d_{2} is adapted to 𝒮 if and only if d_{2} is equivalent to d_{1}.

By the above results, if there exists a qdistance which is adapted to 𝒮, then

{m|there exists a qdistance which is m-adapted to 𝒮}

={m|δ^{(m)} is a quasidistance} = {n, n + 1, …}.

Denote this n by n_{A}(𝒮). Combining Theorems 2.2.6 and Corollary 2.3.12, we have the following result on existence of an adapted qdistance for an intersection type finite scale.

Theorem 2.3.13. Let 𝒮 be a scale on Σ with (EL1). If 𝒮 is intersection type finite with respect to L, then there exists a qdistance on K which is adapted to 𝒮. Furthermore, n_{A}(𝒮) ≤ 3.

Proof. By Theorem 2.2.6, there exists α > 0 such that D_{𝒮^{α}} is a distance on K which is 3-adapted to 𝒮^{α}. Therefore, if d = (D_{𝒮^{α}})^{1/α}, then d is a qdistance on K which is 3-adapted to 𝒮. □

If the self-similar structure is strongly finite, then we have slightly better result.

Theorem 2.3.14. Assume that the self-similar structure L is strongly finite. If 𝒮 is intersection type finite and satisfies (EL1), then δ^{(1)} is a quasidistance. In particular, n_{A}(𝒮) = 1.

Proof. Let (s, x) ∈ (0, 1] × K. For any k ≥ 0 and m ≥ 2, define

CH(x, s, k, m) = {(w(1)v(1), …, w(m)v(m)) ∈ CH|w(i) ∈ Λ_{s} and

v(i) ∈ W_{k} for any i = 1, …, m, x ∈ K_{w(1)v(1)}}.

Also define

K_{m}(s, x, k) = ⋃_{(τ(1),...,τ(m))∈CH(s,x,k,m)} ( ⋃_{i=1}^{m} K_{τ(i)})

Let d be a distance on K which gives the original topology of K. Then the diameter of K_{m}(s, x, k) with respect to d converges to 0 as k → ∞. Since U_{s}(x) is a neighborhood of x, there exists k_{0} such that K_{m}(s, x, k_{0}) ⊆ U_{s}(x). Since 𝒮 satisfies (EL1), there exists α_{1} ∈ (0, 1) such that Λ_{s} ∩ Λ_{α_{1}s} = ∅. This means that any w ∈ Λ_{α_{1}s} can be written as w = w′v, where w′ ∈ Λ_{s} and |v| ≥ 1. Hence if β = (α_{1})^{k_{0}}, then any w ∈ Λ_{βs} can be written as w = w′v, where w′ ∈ Λ_{s} and |v| ≥ k. This along with that fact that K_{m}(s, x, k_{0}) ⊆ U_{s}(x) yields that U_{βs}^{m−1}(x) ⊆ U_{s}(x). Note that the constant β is determined by (s, x) and m. In this sense, we write β = β(s, x, m).

By Theorem 2.2.13, ((0, 1] × K)/∼_{1} is a finite set. Suppose that (s_{1}, x_{1}) ∼_{1}

(s_{2}, x_{2}). Then there exists an L-isomorphism ψ between Λ_{s_{1},x_{1}}^{1} and Λ_{s_{2},x_{2}}^{1}. Using ψ, we see that β(s_{1}, x_{1}, m) = β(s_{2}, x_{2}, m). Since the equivalence class under ∼_{1} is finite,

we may choose β_{1} ∈ (0, 1) such that U_{β_{1}s}^{(m−1)}(x) ⊆ U_{s}^{(n)}(x) for any (s, x) ∈ (0, 1]×K. This implies that δ^{(m−1)}(x, y) ≥ β_{1}δ^{(n)}(x, y) for any x, y ∈ K. In the case, m = 4, we have the condition (F) of Theorem 2.3.11 with n = 1. Therefore δ^{(1)} is a quasidistance. □

In the case of a rationally ramified self-similar structure, Corollary 2.2.8 along with Theorem 2.2.6 implies the following result.

Corollary 2.3.15. Let L be a rationally ramified self-similar structure and let 𝒮 be an elliptic scale on 𝒮. If 𝒮 is locally finite, then n_{A}(𝒮) = 1 and there exists a qdistance on K which is 1-adapted to 𝒮. In particular, if L is post critically finite, then there exists an adapted qdistance for every elliptic scale on Σ.

For self-similar scales, we have the following stronger result.

Theorem 2.3.16. Assume that D_{𝒮(a)} is a distance on K, where a ∈ (0, 1)^{S}. If L is strongly finite and 𝒮(a) is intersection type finite, then there exists β_{1} > 0 such that

B_{β_{1}s}(x, D_{𝒮(a)}) ⊆ U_{s}(x) ⊆ B_{2s}(x, D_{𝒮(a)})

for any s ∈ (0, 1] and any x ∈ K.

Proof. By Proposition 2.1.4, we have U_{s}(x) ⊆ B_{2s}(x, D_{𝒮(a)}). Hereafter we write 𝒮 = 𝒮(a). Let a = (a_{i})_{i∈S} and define c = min_{i∈S} a_{i}. Let X be a finite subset of W_{∗}. If ∪_{w∈X}K_{w} is connected, we define, for x, y ∈ ∪_{w∈X}K_{w},

CH(x, y : X) = {(w(1)v(1), …, w(m)v(m)) ∈ CH(x, y)|w(1), …, w(m) ∈ X}.

and D_{𝒮,X} (x, y) = inf{∑_{j=1}^{m} a_{τ(j)}|(τ(1), …, τ(m)) ∈ CH(x, y : X)}. Note that D_{𝒮,X} (x, y) ≥ D_{𝒮}(x, y). Also for (s, x) ∈ (0, 1] × K, we define

d_{s,x} = inf{D_{𝒮,Λ_{s,x}^{2}} (x_{1}, x_{2})|x_{1} ∈ K_{s}(x), x_{2} ∈ U_{s}^{(2)}(x)\U_{s}(x)}
D_{s,x} = inf{D_{𝒮}(x_{1}, x_{2})|x_{1} ∈ K_{s}(x), x_{2} ∈ U_{s}^{(2)}(x)\U_{s}(x)}.

By Theorem 2.2.13, ((0, 1]×K)/∼_{2} is a finite set. Choose one representative (s_{∗}, x_{∗})

in a equivalence class. Suppose that (s, x) ∼_{2} (s_{∗}, x_{∗}). Let ψ be an 2-isomorphism

between (s, x) and (s_{∗}, x_{∗}) and let ϕ be the homeomorphism between U_{s}^{(2)}(x) and U_{s_{∗}}^{(2)}(x_{∗}) associated with ψ. For p, q ∈ U_{s}^{(2)}(x),

(s_{∗})^{−1} ∑_{j=1}^{m} a_{ψ(w(j))v(j)} ≤ ∑_{j=1}^{m} a_{v(j)} ≤ (cs)^{−1} ∑_{j=1}^{m} a_{w(j)v(j)}

for any (w(1)v(1), …, w(m)v(m)) ∈ CH(p, q : Λ_{s,x}^{2}), where w(1), …, w(m) ∈ Λ_{s,x}^{2}. Hence we have csD_{𝒮,Λ_{s∗,x∗}^{2}} (ϕ(p), ϕ(q))/s_{∗} ≤ D_{𝒮,Λ_{s,x}^{2}} (p, q). This implies that d_{s,x} ≥ c_{∗}s, where c_{∗} = c(s_{∗})^{−1}d_{s_{∗},x_{∗}}. Since the number of equivalence classes is finite, there exists β > 0 such that d_{s,x} ≥ βs for any (s, x) ∈ (0, 1] × K.

If d_{s,x} > D_{s,x}, then there exists a chain between x and y which gives the infimum of the definition of D_{s,x}. This chain should contain a word in Λ_{s′} for s′ ≥ s. Therefore D_{s,x} ≥ s. Combining this with the fact that d_{s,x} ≥ βs, we see that B_{β_{1}s}(x, D_{𝒮}) ⊆ U_{s}(x), where β_{1} = β/2. □

Finally we define the notion of “volume doubling with respect to a qdistance” and consider measures which have volume doubling property.

Theorem 2.3.17. Let L be a rationally ramified self-similar structure and let 𝒮 be an elliptic scale on Σ. Also let μ ∈ M(K). Then μ has the volume doubling property with respect to 𝒮 (i.e. (VD) is satisfied) if and only if the following condition (VDd) is satisfied:

(VDd) There exist a qdistance d on K which is adapted to 𝒮, α ∈ (0, 1) and c > 0 such that μ(B_{s}(x, d)) ≤ cμ(B_{αs}(x, d)) for any s ∈ (0, 1] and any x ∈ K.

Proof. If (VD) holds, then 𝒮 is locally finite by Theorem 1.3.5. Hence by Corollary 2.3.15, there exists a qdistance on K which is adapted to 𝒮. Now (VD) immediately implies (VDd). Conversely (VDd) implies (VD)_{n} for some n. Hence we obtain (VD). □

CHAPTER 3 Heat Kernel and Volume Doubling Property of Measures

3.1. Dirichlet forms on self-similar sets

We now begin to study heat kernels derived from “self-similar” Dirichlet forms on self-similar sets. More precisely, we will establish an equivalence between certain type of upper heat kernel estimate and the volume doubling property. See the next section for details. In this section, we will give a framework on “self-similar” Dirichlet forms. Let L = (K, S, {F_{i}}_{i∈S}) be a self-similar structure. Hereafter we will always assume that K ≠ \overline{V}_{0} and that K is connected.

The following lemma is easy to verify.

Lemma 3.1.1. Let μ be an elliptic probability measure on K. Then, for any w ∈ W_{∗}, there exists a unique elliptic probability measure μ^{w} on K such that μ^{w}(A) = μ(F_{w}(A))/μ(K_{w}) for any Borel set A ⊆ K. Moreover, define ρ_{w} : L^{2}(K, μ) → L^{2}(K, μ^{w}) by ρ_{w}u = u∘F_{w}. Then ρ_{w} is a bounded operator.

Remark. If μ is a self-similar measure on K with weight (μ_{i})_{i∈S}, then μ^{w} = μ for any w ∈ W_{∗}

Now we define the notion of self-similar Dirichlet forms.

Definition 3.1.2. Let μ be an elliptic probability measure on K and let (E, F) be a local regular Dirichlet form on L^{2}(K, μ).

(1) We say that (E, F, μ) is self-similar, (SSF) for short, if and only if it satisfies the following two conditions: (SSF1) u ∘ F_{i} ∈ F for any i ∈ S and any u ∈ F. There exists (r_{i})_{i∈S} ∈ (0, ∞)^{S} such that

(3.1.1)  E(u, v) = ∑_{i∈S} \frac{1}{r_{i}} E(u∘F_{i}, v∘F_{i})

for any u, v ∈ F. If g(w) = \sqrt{r_{w}μ(K_{w})}, then g(w) is a gauge function and the scale 𝒮_{∗} induced by g is elliptic. (SSF2) Let Γ_{1} and Γ_{2} be subsets of W_{∗} which are L-similar and let ψ be the associated L-similitude between K(Γ_{1}) and K(Γ_{2}). If u ∈ F, supp(u) ⊆ K(Γ_{1}) and u∘ψ|_{∂K(Γ_{2})} ≡ 0, then there exists v ∈ F such that supp(v) ⊆ K(Γ_{2}) and v|_{K(Γ_{2})} = u∘ψ.

The ratio (r_{i})_{i∈S} is called the resistance scaling ratio. If r_{i} < 1 for any i ∈ S, then (E, F, μ) is said to be recurrent.

Remark. (1) If μ is a self-similar measure with weight (μ_{i})_{i∈S}, then g(w) is a gauge function if and only if r_{i}μ_{i} < 1 for any i ∈ S. In this case 𝒮_{∗} is always elliptic.

(2) If (E, F, μ) is recurrent, then g(w) is always a gauge function and 𝒮_{∗} is always elliptic.

Definition 3.1.3. Let μ be an elliptic probability measure on K and let (E, F) be a local regular Dirichlet form on L^{2}(K, μ). We say that (E, F, μ) satisfy Poincaré inequality, (PI) for short, if and only if there exists c > 0 such that

(PI)  E(u, u) ≥ c ∫_{K}(u − (ū)_{μ^{w}}))^{2}dμ^{w}

for any w ∈ W_{∗} and any u ∈ ρ_{w}(F), where (ū)_{ν} = ∫_{K} udν.

Remark. If μ is a self-similar measure, then μ^{w} = μ for any w ∈ W_{∗}. Therefore, in this case, (PI) holds if and only if

E(u, u) ≥ c ∫_{K}(u − ū)^{2}dμ

for any u ∈ F, where c is a positive constant. Furthermore assume that (E, F) is conservative, i.e. 1 ∈ F and E(1, 1) = 0. Let −Δ be the non-negative selfadjoint operator associated with the Dirichlet form (E, F) on L^{2}(K, μ). Then by the variational principle, (PI) holds if and only if 0 is the eigenvalue of H whose multiplicity is one and the spectrum of −Δ is contained in {0} ∪ [c, ∞) for some c > 0.

Hereafter we always assume that μ is an elliptic probability measure on (K, d) and that (E, F) is a local regular Dirichlet form on L^{2}(K, μ). From the self-similarity (SSF) and the Poincaré inequality (PI), we can establish the existence of heat kernels and their diagonal estimates.

Theorem 3.1.4. Assume that (E, F, μ) satisfy the conditions (SSF) and (PI). Let {T_{t}}_{t>0} be the strongly continuous semigroup on L^{2}(K, μ) associated with the Dirichlet form (E, F). Then {T_{t}}_{t>0} is ultracontractive and there exist α > 0 and c > 0 such that ||T_{t}||_{1→∞} ≤ ct^{−α/2} for any t ∈ (0, 1]. Moreover, there exists p : (0, +∞) × K × K → [0, +∞) such that p(t, ·, ·) ∈ L^{∞}(K × K) and

(T_{t}u)(x) = ∫_{K} p(t, x, y)u(y)μ(dy)

for any u ∈ L^{2}(K, μ). p(t, x, y) is called the heat kernel associated with the Dirichlet form (E, F) on L^{2}(K, μ). In particular, if (E, F, μ) is recurrent, then α ∈ (0, 2).

We need the next two lemmas to show the above theorem.

Lemma 3.1.5. Let Λ be a partition of Σ. For any u ∈ F, define Λ(u) = {w|w ∈ Λ, K_{w} ∩supp(u) ≠ ∅}. Assume that (E, F, μ) satisfy the conditions (SSF) and (PI). Then

E(u, u) + \frac{c}{min_{w∈Λ(u)} r_{w}μ(K_{w})^{2}} ||u||_{1}^{2} ≥ \frac{c}{max_{w∈Λ(u)} r_{w}μ(K_{w})}||u||_{2}^{2},

where c is the constant appearing in (PI).

Proof. Using (3.1.1) and (PI), we see that

E(u, u) = ∑_{w∈Λ(u)} \frac{1}{r_{w}} E(u ∘ F_{w}, u ∘ F_{w})

≥ ∑_{w∈Λ(u)} \frac{c}{r_{w}} ( ∫_{K}(u ∘ F_{w})^{2}dμ^{w} − ( ∫_{K} u ∘ F_{w}dμ^{w})^{2})

≥ ∑_{w∈Λ(u)} \frac{c}{r_{w}μ(K_{w})} ∫_{K_{w}} u^{2}dμ − ∑_{w∈Λ(u)} \frac{c}{r_{w}μ(K_{w})^{2}} ( ∫_{K_{w}} udμ)^{2}

≥ \frac{c}{max_{w∈Λ(u)} r_{w}μ(K_{w})}||u||_{2}^{2} − \frac{c}{min_{w∈Λ(u)} r_{w}μ(K_{w})^{2}} ||u||_{1}^{2}.

□

Lemma 3.1.6. Assume that (E, F, μ) satisfy the conditions (SSF) and (PI). Let 𝒮_{∗} = {Λ_{s}}_{s∈(0,1]}. Then there exist positive constants c_{1} and c_{2} such that

(3.1.2)  E(u, u) + \frac{c_{1}}{s^{2} min_{w∈Λ_{s}(u)} μ(K_{w})}||u||_{1}^{2} ≥ \frac{c_{2}}{s^{2}} ||u||_{2}^{2}

for any u ∈ F and any s ∈ (0, 1].

Proof. Since μ is elliptic, μ(K_{w_{1}...w_{m}}) ≥ αμ(K_{w_{1}...w_{m−1}}) for any w_{1} … w_{m} ∈ W_{∗}, where α > 0 is independent of w. Hence for any w ∈ Λ_{s}, it follows that αrs^{2} ≤ g(w) ≤ s^{2}, where r = min_{i∈S} r_{i}. This along with Lemma 3.1.5 immediately implies (3.1.2). □

Proof of Theorem 3.1.4. Since μ and 𝒮_{∗} is elliptic, there exist δ, η ∈ (0, 1), c_{1} > 0 and c_{2} > 0 such that μ(K_{w}) ≥ c_{1}δ^{|w|} and g(w) ≤ c_{2}η^{|w|} for any w ∈ W_{∗}. Therefore, there exist positive constants α and c_{3} such that μ(K_{w}) ≥ c_{3}g(w)^{α} for any w ∈ W_{∗}. This with (3.1.2) implies that

(3.1.3)  E(u, u) + \frac{c_{4}}{s^{2+α}} ||u||_{1}^{2} ≥ \frac{c_{5}}{s^{2}} ||u||_{2}^{2}

for any u ∈ F ∩ L^{1}(K, μ). By [30, Theorem 3.2], (3.1.3) turn out to be equivalent to the Nash inequality (A.1). Using Theorem A.2, we deduce that {T_{t}}_{t>0} is ultracontractive and ||T_{t}||_{1→∞} ≤ ct^{−α/2}. The existence of the heat kernel follows from Theorems A.2 and A.3.

If (E, F, μ) is recurrent, there exist c > 0 and γ > 0 such that μ(K_{w}) ≥ c(r_{w})^{γ} for any w ∈ W_{∗}. Choose α so that γ = (α/2)/(1 − (α/2)). Then α ∈ (0, 2) and μ(K_{w}) ≥ c_{2}g(w)^{α} for any w ∈ W_{∗}. □

We also need the following two properties to establish a suitable framework for heat kernel estimate.

Definition 3.1.7. Assume that (E, F, μ) satisfy the conditions (SSF) and (PI).

(1) (E, F, μ) is said to have the continuous heat kernel, (CHK) for short, if and if (CHK) The heat kernel p(t, x, y) associated with the Dirichlet form (E, F) on L^{2}(K, μ) is jointly continuous, i.e. p : (0, +∞) × K × K → [0, +∞) is continuous.

(2) Let (Ω, {X_{t}}_{t≥0}, {P_{x}}_{x∈K}) be the diffusion process associated with the local regular Dirichlet form (E, F) on L^{2}(K, μ). For any A ⊆ K, we define the hitting time of A, h_{A}, by h_{A} = inf{t ≥ 0|X_{t} ∈ A}. (E, F, μ) is said to have uniform positivity of hitting time, (UPH) for short, if and only if (UPH) inf_{x∈B} E_{x}(h_{A}) > 0 for all closed sets A and B with A ∩ B = ∅.

In the subsequent sections, we will study heat kernels associated with a local regular Dirichlet form (E, F) on L^{2}(K, μ) which satisfy (SSF), (PI), (CHK) and (UPH). A similar set of assumptions on Dirichlet forms on self-similar sets has given in [8, Assumption 2.3]

In the recurrent case, (SSF) along with (PI) implies (CHK) and (UPH).

Theorem 3.1.8. Assume (SSF) and (PI). If (E, F, μ) is recurrent, then (CHK) and (UPH) are satisfied.

Lemma 3.1.9. Assume (SSF), (PI) and that (E, F, μ) is recurrent. Then F ⊆ C(K, d). Let U be an open subset of K. Define F_{U} = {u|u ∈ F, u|_{K\U} = 0} and E_{U} = E|_{F_{U} ×F_{U}}. Also let μ|_{U} be the Borel regular measure on U defined by μ|_{U}(A) = μ(A) for any Borel subset A of U. Then (E_{U}, F_{U}) is a local regular Dirichlet form on L^{2}(U, μ|_{U}). The associated semigroup, {T_{t}^{U}}_{t>0}, on L^{2}(U, μ|_{U}) is ultracontractive and the associated heat kernel p_{U} : (0, +∞) × K × K → [0, +∞) is continuous.

The heat kernel p_{U} itself is only defined on (0, +∞) × U × U by definition. However, we can extend p_{U}(t, x, y) by letting p_{U}(t, x, y) = 0 if x or y belongs to K\U.

Proof. By Theorem 3.1.4, it follows that ||T_{t}||_{1→∞} < ct^{α/2}, where α ∈ (0, 2). Hence applying Theorem A.6, we obtain that F ⊆ C(K, d). Then (E_{U}, F_{U}) is a local regular Dirichlet form on L^{2}(U, μ|_{U}) by [15, Theorem 4.4.3]. Starting from (3.1.3), we follow the same discussion as in the proof of Theorem 3.1.4 and obtain ||T_{t}^{U}||_{1→∞} ≤ ct^{−α/2}. Hence Theorem A.6 shows that p_{U} is continuous. □

Proof of Theorem 3.1.8. We already verify (CHK) in Lemma 3.1.9. Let A be a non-empty closed subset of K. For x ∈ K,

(3.1.4)  E_{x}(h_{A}) = ∫_{0}^{∞} ∫_{X} p_{Y} (t, x, y)μ(dy)dt,

where Y = A^{c}. Since A ≠ ∅, E_{Y} (u, u) = 0 if and only if u = 0. Therefore, if λ_{1} is the smallest eigenvalue of the non-negative self-adjoint operator associated with the Dirichlet form (E_{Y} , F_{Y}) on L^{2}(Y, μ|_{Y}), −Δ_{Y}, then λ_{1} > 0. By (A.2), there exists c_{1} > 0 such that

p_{Y} (t, x, y) ≤ c_{1}e^{−λt}

for any x, y ∈ K and any t ≥ 1. For t ∈ (0, 1], By Lemma 3.1.9, there exists c_{2} > 0 such that p(t, x, y) ≤ c_{2}t^{−α/2} for any x, y ∈ K and t ∈ (0, 1]. Therefore, define

F (t) = \begin{cases}c_{2}t^{−α/2} & if t ∈ (0, 1], \\ c_{1}e^{−λ_{1}t} & if t > 1. \end{cases}

Then p_{Y} (t, x, y) ≤ F (t) and ∫_{0}^{1} ∫_{X} F (t)μ(dy)dt < +∞. Note that p_{Y} (t, x, y) is continuous. By the Lebesgue dominated convergence theorem, (3.1.4) implies that E_{x}(h_{A}) is continuous with respect to x ∈ K. Assume that B is a closed subset of K and A ∩ B = ∅. Since the process is a diffusion process, P_{x}(h_{A} = 0) > 0 for any x ∈ B. Hence E_{x}(h_{A}) > 0 for any x ∈ B. Therefore, inf_{x∈B} E_{x}(h_{A}) = min_{x∈B} E_{x}(h_{A}) > 0. Thus we obtain (UPH). □

Proposition 3.1.10. Assume (SSF), (PI) and (CHK). If (E.F) is conservative, i.e. 1 ∈ F and E(1, 1) = 0, then the heat kernel p(t, x, y) is positive, i.e. p(t, x, y) > 0 for any (t, x, y) ∈ (0, +∞) × K × K.

Proof. By [28, Theorem 1.6.2], K is arcwise connected. Hence the desired result follows from Theorem A.4. □

3.2. Heat kernel estimate

In this section, we will give our main result on heat kernels associated with selfsimilar Dirichlet forms on self-similar sets. Let L = (K, S, {F_{i}}_{i∈S}) be a self-similar structure. Hereafter we will always assume that K ≠ \overline{V}_{0}, that K is connected and that (K, S, {F_{i}}_{i∈S}) is rationally ramified with a relation set R. Moreover, μ is an elliptic probability measure on K and (E, F) is a local regular Dirichlet form on L^{2}(K, μ).

Definition 3.2.1. Assume that (E, F, μ) satisfy (SSF). The resistance scaling ratio (r_{i})_{i∈S} of (E, F) is said to be arithmetic on R_{1}-relations if and only if log r_{w}/ log r_{v} ∈ ℚ for any ({w}, {v}, φ, x, y) ∈ R_{1}.

For the lower off-diagonal estimate of heat kernels, we need a “geodesic” between a pair of points.

Definition 3.2.2. Let (X, d) be a metric space. For x, y ∈ X, a curve γ : [0, d(x, y)] → X is called a geodesic between x and y if and only if γ(0) = x, γ(1) = y and d(γ(t), γ(s)) = |t − s| for any t, s ∈ [0, d(x, y)]. We call (x, y) ∈ X^{2} a geodesics pair for (X, d) if and only if there exists a geodesic between x and y. The distance d is called a geodesic distance if and only if every pair (x, y) ∈ X^{2} is a geodesic pair.

Theorem 3.2.3. Assume that (E, F) is conservative and that (E, F, μ) satisfy (SSF), (PI), (CHK) and (UPH). Let 𝒮_{∗} be the scale induced by the gauge function g(w) = \sqrt{r_{w}μ(K_{w})}. Suppose either that (I) (E, F) is recurrent or that (II) μ is a self-similar measure on K and the resistance scaling ratio (r_{i})_{i∈S} is arithmetic on R_{1}-relations. Then, the following four conditions (a) - (d) are equivalent.

(a) μ is volume doubling with respect to the scale 𝒮_{∗}.

(b) There exists a qdistance d on K adapted to 𝒮_{∗} such that μ is volume doubling with respect to the qdistance d.

(c) There exist c > 0 such that

(DUHK’)  p(t, x, x) ≤ \frac{c}{μ(U\sqrt{t}(x))}

for any t ∈ (0, 1] and any x ∈ K.

(d) There exist a qdistance d on K which is adapted to 𝒮_{∗} and c > 0 such that

(DUHK)  p(t, x, x) ≤ \frac{c}{μ(B\sqrt{t}(x, d))}

for any t ∈ (0, 1] and any x ∈ K.

Moreover, suppose that any of the above conditions holds. Let d be a qdistance adapted to 𝒮_{∗}. If d^{α} is a distance on K, then α < 2 and there exist positive constants c_{1}, c_{2} and c_{3} such that, for any t ∈ (0, 1] and any x, y ∈ K,

(DLHK)  \frac{c_{1}}{μ(B\sqrt{t}(x, d))} ≤ p(t, x, x)

and

(UHK)  p(t, x, y) ≤ \frac{c_{2}}{μ(B\sqrt{t}(x, d))} exp ( − c_{3}(\frac{d(x, y)^{2}}{t} )^{\frac{1}{β−1}} ),

where β = 2/α. Also in the recurrent case, there exist positive constants c_{4} and c_{5} such that

(LHK)  \frac{c_{4}}{μ(B\sqrt{t}(x, d))} exp ( − c_{5}(\frac{d(x, y)^{2}}{t} )^{\frac{1}{β−1}} ) ≤ p(t, x, y)

for any t ∈ (0, 1] and any geodesic pair (x, y) ∈ K^{2} for (K, d^{α}).

Remark. At a glance, it seems that the inequalities (DUHK), (DLHK) and (UHK) may depend on the choice of a qdistance d. Using U_{s}(x) and δ^{(1)}(x, y), however, we may rewrite those inequalities. Namely, if δ^{(1)}(x, y)^{α} is equivalent to a distance on K, then

\frac{γ_{1}}{μ(U\sqrt{t}(x))} ≤ p(t, x, x) ≤ \frac{γ_{2}}{μ(U\sqrt{t}(x))}

and

p(t, x, y) ≤ \frac{γ_{2}}{μ(U\sqrt{t}(x))} exp ( − γ_{3}(\frac{δ^{(1)}(x, y)^{2}}{t} )^{\frac{1}{β−1}} ),

where β = 2/α. Note that γ_{1} and γ_{2} are independent of α. The constant γ_{3} is the only place where the value of α may be involved.

We will give a proof of Theorem 3.2.3 in Section 3.5.

There are two classes of self-similar sets, p. c. f self-similar sets and Sierpinski carpets, where a local regular Dirichlet form with (SSF), (PI), (CHK) and (UPH) has been constructed. We will apply the above theorem to those classes in the next two sections.

3.3. P. c. f. self-similar sets

In this section, we will consider post critically finite self-similar structures. In this case, one can easily determine when the assumptions of Theorem 3.2.3 hold. Throughout this section, L = (K, S, {F_{i}}_{i∈S}) is a post critically finite self-similar structure whose relation set is {({w(i)}, {v(i)}, φ_{i}, x(i), y(i))|i = 1, …, m}, where w(i), v(i), x(i), y(i) ∈ W_{#} and φ_{i}(w(i)) = v(i).

There is an established way of constructing self-similar Dirichlet forms on a post critically finite self-similar sets in [28]. It starts from a harmonic structure (D, r), where D is a “Laplacian” on V_{0}, which is a finite set for a p. c. f. self-similar set, and r = (r_{i})_{i∈S} ∈ (0, ∞)^{S}. From (D, r), we obtain a quadratic form (E, F) which satisfies u ∘ F_{i} ∈ F for any i ∈ S and

E(u, u) = ∑_{i∈S} \frac{1}{r_{i}} E(u ∘ F_{i}, u ∘ F_{i})

for any u ∈ F. See [28] for details.

We assume that μ is an elliptic probability measure on K for the rest of this section.

Proposition 3.3.1. Assume either that (D, r) is recurrent, i.e. r ∈ (0, 1)^{S}, or that μ is a self-similar measure with weight (μ_{i})_{i∈S} which satisfies r_{i}μ_{i} < 1 for any i ∈ S. Then (E, F) is an local regular Dirichlet form on L^{2}(K, μ) which satisfies (SSF), (PI), (CHK) and (UPH).

Proof. If (D, r) is recurrent, then the conditions (RFA1), (RFA2) and (RFA3) are immediately verified. Hence the statement follows by Theorem B.3. Next assume that μ is a self-similar measure with weight (μ_{i})_{i∈S} which satisfies r_{i}μ_{i} < 1 for any i ∈ S. Then we have (SSF) by the method of construction of F. See [28, Sections 3.1 and 3.2] for details. Also by [28, Theorem 3.4.6], (E, F) is a local regular Dirichlet form on L^{2}(K, μ) and the associated non-negative self-adjoint operator H has compact resolvent. Also the kernel of H is equal to constants. Therefore, by the remark after Definition 3.1.3, we obtain (PHI). By [28, Prposition 5.1.2], we also have (CHK). Finally, we show (UPH). Let A and B be closed subsets of K with A ∩ B = ∅. Set A_{m} = K(W (W_{m}, A)). Then A_{m} ∩ B = ∅ for sufficiently large m. Since A ⊆ A_{m}, we have E_{x}(h_{A_{m}}) ≤ E_{x}(h_{A}). Therefore, we may replace A by A_{m} to show (UPH). In other word, we may regard A as ∪_{w∈Γ}K_{w} for some finite subset Γ of W_{∗}. In such a case, ∂A is a finite subset of V_{∗} and h_{A} = h_{∂A} for any path starting from B. By [28, Section A.2], the heat kernel p_{∂A}(t, x, y) corresponding to the Dirichlet form (E, F_{∂A}) on L^{2}(K, μ) is jointly continuous on (0, ∞) × K^{2}. Also, we have

E_{x}(h_{A}) = ∫_{K\A} ∫_{0}^{∞} p_{∂A}(t, x, y)dtμ(dy)

for any x ∈ K\A. Define

F (x) = ∫_{K\A} ∫_{0}^{∞} p_{∂A}(t, x, y)dtμ(dy).

By definition, 0 ≤ F (x) ≤ E_{x}(h_{A}) for any x ∈ B. By [28, Theorem A.2.1], the nonnegative self-adjoint operator −Δ_{∂A} associated with (E, F_{∂A}) on L^{2}(K, μ) has compact resolvent. Let λ_{∗} be the smallest eigenvalue of −Δ_{∂A}. If E(u, u) = 0, then u is constant and u|_{∂A} ≡ 0. This implies that λ_{∗} > 0. Hence there exists C > 0 such that

p_{∂A}(t, x, y) ≤ Ce^{−λ_{∗}t}

for any (t, x, y) ∈ [1, ∞) × K^{2}. Hence F (x) is continuous on K\A by the Lebesgue dominated convergence theorem. Moreover by [28, Theorem A.2.19], we have p_{∂A}(t, x, x) > 0 for any x ∈ K\A. Hence F (x) > 0 for any x ∈ K\A. Since B is compact, we deduce that 0 < inf_{x∈B} F (x) ≤ inf_{x∈B} E_{x}(h_{A}). Thus we obtain (UPH). □

From now on, we confine ourselves to the second case in the above proposition, namely, μ is a self-similar measure with weight (μ_{i})_{i∈S} which satisfies r_{i}μ_{i} < 1 for any i ∈ S. Note that if (D, r) is recurrent, then the assumption (I) of Theorem 3.2.3 is satisfied. If not, the resistance scaling ratio r should be arithmetic on R_{1}-relations in order to satisfy the assumption (II) of Theorem 3.2. Note that every relation is an R_{1}-relation for p. c. f. self-similar structure.

Proposition 3.3.2. The assumption (II) of Theorem 3.2.3 holds if and only if log r_{w(i)}/ log r_{v(i)} ∈ ℚ for any i = 1, …, m.

Proof. This is immediate by Definition 3.2.1. □

We have the following simple condition which is equivalent to the statement (a) of Theorem 3.2.3

Proposition 3.3.3. Let 𝒮_{∗} be the scale induces by the gauge function g(w) = \sqrt{r_{w}μ_{w}}. Then μ has the volume doubling property with respect to 𝒮_{∗} if and only if

\begin{matrix}log & r_{w(i)} \\ log & μ_{w(i)} \end{matrix} = \begin{matrix}log & r_{v(i)} \\ log & μ_{v(i)} \end{matrix}

for any i = 1, …, m.

Proof. Corollary 1.6.13 suffices to show the desired statement. □

If μ has the volume doubling property with respect to 𝒮_{∗}, we can apply Theorem 3.2.3 and obtain heat kernel estimates. As is seen in the last section, if

(3.3.1)  max{α|D_{γ^{α}} is a distance on K}

exists, then it plays an important role in off-diagonal heat kernel estimates like (UHK) and (LHK). Next we study how to calculate the value of maximum in (3.3.1).

Definition 3.3.4. (1) Define

CH_{m}(x, y) = {(w(j))_{j=1,...,k}|(w(j))_{j=1,...,k} ∈ CH(x, y),

w(j) ∈ W_{m} for any j = 1, …, k},

for x, y ∈ K and m ≥ 0. We regard CH_{1}(x, y) as a subset of W_{#} by identifying (w(j))_{j=1,...,k} ∈ CH_{1}(x, y) with w(1)w(2) … w(k) ∈ W_{#}.

(2) (A, τ) is called a recursive system of paths if A is a non-empty finite subset of

⋃_{p,q∈V_{0}:p≠q}{(w, p, q), w ∈ CH_{1}(p, q)},

and τ : A → ∪_{n≥1}A^{n} satisfies the following condition: τ((w, p, q)) ∈ A^{|w|} for any (w, p, q) ∈ A. If τ((w_{1} … w_{k}, p, q)) = ((w^{(j)}, p_{j}, q_{j}))_{j=1,...,k}, then p = F_{w_{1}}(p_{1}), q = F_{w_{k}}(q_{k}) and F_{w_{j}}(q_{j}) = F_{w_{j+1}}(p_{j+1}) for any j = 1, …, k − 1.

(3) Let (A, τ) be a recursive system of paths. (A, τ) is called irreducible if and only if B = A whenever B ⊆ A and (B, τ|_{B}) is a recursive recursive system of paths.

(4) Let (A, τ) be recursive and let a = (a_{j})_{j∈S} ∈ (0, 1)^{S}. Then the relation matrix M = M_{A,τ,a} is a #A × #A-matrix defined by

M_{w,w′} = ∑_{j:w^{(j)}=w′} a_{w_{j}}

where w = (w_{1} … w_{k}, p, q) and τ(w) = ((w^{(1)}, …, w^{(k)})).

In some cases, the following results are useful in determining whether D_{a} is a distance or not. In fact, later in this section, we will make use of them to characterize the value (3.3.1) for an example.

Proposition 3.3.5. Let a = (a_{j})_{j∈S} ∈ (0, 1)^{S}.

(1) If ∑_{j=1}^{k} a_{w_{j}} ≥ 1 for any w_{1} … w_{k} ∈ ∪_{p,q∈V_{0}:p≠q}CH_{1}(p, q), then D_{a} is a distance on K.

(2) If there exists a recursive (A, τ) such that the maximum eigenvalue of the relation matrix M_{A,τ,a} is less than one, then D_{a} is not a distance.

The following notations are convenient in proving the above proposition.

Notation. Let w = (w(j))_{j=1,...,k} ∈ CH(x, y).

(1) For v = (v(j))_{j=1,...,l} ∈ CH(y, z), we use w ∨ v to denote the chain between x and z defined by ((w(1), …, w(k), v(1), …, v(l)). In the same manner, we define ∨_{i=1}^{n}w_{i} ∈ CH(x_{1}, x_{n}) if w_{i} ∈ CH(x_{i}, x_{i+1}) i = 1, …, n − 1.

(2) For v ∈ W_{∗}, define vw = (vw(j))_{j=1,...,k}, which is a chain between F_{v}(x) and F_{v}(y).

(3) For a = (a_{j})_{j∈S} ∈ (0, 1)^{S} and w = (w(j))_{j=1,...,k} ∈ CH, define

a_{w} = ∑_{j=1}^{k} a_{w(j)}.

Proof. (1) Assume that p, q ∈ V_{0} and p ≠ q. Let (w(j))_{j=1,...,n} ∈ CH(p, q). We will show that

(3.3.2)  ∑_{j=1}^{n} a_{w(j)} ≥ 1

by using induction on n. If n = 1, then w(1) = ∅. Since a_{∅} = 1, (3.3.2) holds. Also if (w(j))_{j=1,...,n} ∈ CH_{1}(p, q), then (3.3.2) also holds by the assumption of the proposition. Otherwise, there exist w ∈ W_{#}, j_{∗} ∈ {1, …, n} and i_{1}, …, i_{l} ∈ S such that w(j_{∗} + k − 1) = wi_{k} for k = 1, …, l and (i_{k})_{k=1,...,l} ∈ CH_{1}(p′, q′) for some p′, q′ ∈ V_{0} with p′ ≠ q′. Let (w′(1), …, w′(n − l + 1)) = (w(1), …, w(j_{∗} − 1), w, w(j_{∗} + l), …, w(n)). Then (w′(1), …, w′(n − l + 1)) ∈ CH(p, q). Using the assumption of the proposition and induction, we obtain

∑_{j=1}^{n} a_{w(j)} = ∑_{j=1}^{j_{∗}−1} a_{w(j)} + a_{w} ∑_{k=1}^{l} a_{i_{k}} + ∑_{j=j_{∗}+l}^{n} a_{w(j)} ≥ ∑_{m=1}^{n−l+1} a_{w′(m)} ≥ 1.

Therefore, (3.3.2) holds for any element of CH(p, q). This immediately implies that D_{a}(p, q) ≥ a_{w} for any w ∈ W_{∗} and any p, q ∈ F_{w}(V_{0}) with p ≠ q.

Next, define K_{m}(x) = ∪_{w∈W_{m}:x∈K_{w}}K_{w}. For any x, y ∈ K with x ≠ y, we may choose m ≥ 1 such that K_{m}(x) ∩ K_{m}(y) = ∅. Then for any (w(j))_{j=1,...,n} ∈ CH(x, y), there exist j_{∗}, l and p, q ∈ V_{m} with p ≠ q such that (w(j_{∗}), …, w(j_{∗} + l − 1)) ∈ CH(p, q). This shows that D_{a}(x, y) ≥ min_{p,q∈V_{m}:p≠q} D_{a}(p, q) > 0. Thus D_{a} is a distance.

(2) Let (A, τ) be a recursive system of paths. First define τ_{m}(w, p, q) ∈ CH_{m}(p, q) for (w, p, q) ∈ A inductively as follows. Set τ_{1}(w, p, q) = w for any (w, p, q) ∈ A. If τ(w, p, q) = ((w(j), p_{j}, q_{j}))_{j=1,...,k}, then we define τ_{m}(w, p, q) = ∨_{j=1}^{k}w_{j}τ_{m−1}(w(j), p_{j}, q_{j}), where w = w_{1} … w_{k}.

Let a = (a_{j})_{j∈S} ∈ (0, 1)^{S}. Then a_{τ_{m}(w,p,q)} = (M^{m}e)_{(w,p,q)} for any (w, p, q) ∈ A, where M = M_{A,τ,a} and e ∈ ℓ(A) is the transpose of (1, …, 1). Assume that the maximum eigenvalue of M is less than one. It follows that the maximum eigenvalue of M_{B,τ|_{B},a} is less than one if B ⊆ A and (B, τ|_{B}) is a recursive system of paths. On the other hand, there exists an irreducible recursive system of paths (B, τ′) where B ⊆ A and τ′ = τ|_{B}. Therefore, A may be assumed to be irreducible without loss of generality. Let λ be the maximum eigenvalue of M Then by the Perron-Frobenius theorem, 0 < λ < 1 and we can choose a positive vector f as an associated eigenvector. Since e ≤ cf for some c > 0, M^{n}e ≤ cλ^{n}f as n → ∞. Hence

p_{1}^{p_{2}}  ^{p_{3}}^{p_{4}}^{p_{6}}  ^{p_{5}} 2  4  3 6  5 1

Figure 3.1. the modified Sierpinski gasket

lim_{m→∞} a_{τ_{m}(w,p,q)} = 0. This implies that D_{a}(p, q) = 0 if (w, p, q) ∈ A. Therefore D_{a} is not a distance. □

Finally, we apply the above results to a particular example.

Definition 3.3.6. Set p_{1} = e^{\sqrt{−1}π/6}, p_{2} = 0, p_{3} = 1, p_{4} = (p_{2} + p_{3})/2, p_{5} = (p_{3} + p_{1})/2 and p_{6} = (p_{1} + p_{2})/2. Define F_{i} : ℂ → ℂ by F_{i}(z) = (z − p_{i})/3 + p_{i} for i = 1, …, 6. Let K be the unique non-empty compact set that satisfies K = ∪_{i∈S}F_{i}(K), where S = {1, …, 6}. K is called the modified Sierpinski gasket.

In the rest of this section, K is assumed to be the modified Sierpinski gasket and L = (K, S, {F_{i}}_{i∈S}) is the associated self-similar structure defined above. Immediately by the above definition, we obtain the following.

Proposition 3.3.7. The relation set of L is

{({1}, {2}, φ_{12}, i, j)|(i, j) = (6, 1), (2, 6), (4, 5)}

∪ {({2}, {3}, φ_{23}, i, j)|(i, j) = (4, 2), (3, 4), (5, 6)}

∪ {({3}, {1}, φ_{31}, i, j)|(i, j) = (1, 5), (5, 3), (6, 4)},

where φ_{kl}(k) = l for (k, l) = (1, 2), (2, 3), (3, 1). In particular, L is post critically finite, P = {(1)^{∞}, (2)^{∞}, (3)^{∞}) and V_{0} = {p_{1}, p_{2}, p_{3}}.

Proposition 3.3.8. Let D = ( \begin{matrix}−2 & 1 & 1 \\ 1 & −2 & 1 \\ 1 & 1 & −2 \end{matrix} ) and let r = (\frac{7}{15}, … \frac{7}{15}).

(1) (D, r) is a recurrent harmonic structure on (K, S, {F_{i}}_{i∈S}).

(2) Let μ be a self-similar measure on K with weight (μ_{i})_{i∈S} and let 𝒮_{∗} be a self-similar scale with weight \{γ_{i}\}_{i∈S}, where γ_{i} = \sqrt{μ_{i}r_{i}}. Then μ has the volume doubling property with respect to 𝒮_{∗} if and only if μ_{1} = μ_{2} = μ_{3}.

Proof. (1) This can be shown by the Δ-Y transform. See [28] for details.

(2) Apply Proposition 3.3.3. □

p_{1}^{p_{2}}  ^{p_{3}}^{p_{4}}^{p_{6}}  ^{p_{5}} μ_{∗}  μ_{4}  μ_{∗} μ_{6}  μ_{5} μ_{∗} μ_{∗} = μ_{1} = μ_{2} = μ_{3}

Figure 3.2. Self-similar volume doubling measures on the modified Sierpinski gasket

Hereafter, we fix (D, r) and 𝒮_{∗} as in the above proposition. Also μ is assumed to be a self-similar measure which satisfies μ_{1} = μ_{2} = μ_{3}. See Figure 3.2. Note that we may assume that μ_{4} ≤ μ_{5} ≤ μ_{6} without loss of generality.

Proposition 3.3.9. Assume that μ_{4} ≤ μ_{5} ≤ μ_{6}. Let α_{∗} be the unique α which satisfies

(3.3.3)  2(γ_{1})^{α} + (γ_{4})^{α} = 1.

Then

α_{∗} = max{α|D_{γ^{α}} is a distance on K},

where γ^{α} = ((γ_{i})^{α})_{i=1,...,6}.

Note that γ_{i} = \sqrt{7μ_{i}/15} for any i.

Proof. Let w = (243, p_{2}, p_{3}). Note that 243 ∈ CH_{1}(p_{2}, p_{3}). Set A = {w} and define τ : A → A^{3} by τ(w) = (w, w, w). Then (A, τ) is a recursive system of paths and M_{A,τ,γ^{α}} = (2(γ_{1})^{α} + (γ_{4})^{α}). If α > α_{∗}, the maximum eigenvalue of M_{A,τ,γ^{α}} is less than one. Hence Proposition 3.3.5-(2) implies that D_{γ^{α}} is not a distance. On the other hand, for α = α_{∗}, we may verify the assumption of Proposition 3.3.5-(1) and show that D_{γ^{α}} is a distance. □

Theorem 3.3.10. Assume that μ_{4}, ≤ μ_{5} ≤ μ_{6}. Let (E, F) be the Dirichlet form associated with (D, r) on L^{2}(K, μ) and let p(t, x, y) be the corresponding heat kernel.Also define d = (D_{γ^{α∗}})^{1/α_{∗}}, where α_{∗} is the unique solution of (3.3.3).

(1) Suppose that μ_{4} < μ_{5}. Then, (UHK) and (DLHK) holds for any x, y ∈ K and any t ∈ (0, 1] with β = β_{∗}. Moreover, (LHK) holds if the line segment \overline{xy} is contained in K and is parallel to the real axis.

(2) If μ_{4} = μ_{5}, then (UHK) and (LHK) holds for any x, y ∈ K and any t ∈ (0, 1] with β = β_{∗}.

Proof. In both cases, Theorem 3.2.3 immediately implies (UHK). Assume that the line segment \overline{xy} is contained in K and is parallel to the real axis. Then we see that (x, y) is a geodesic pair for (K, D_{γ^{α∗}}). Hence by Theorem 3.2.3, we have (LHK) for such a pair. In the case (2), it follows that D_{γ^{α∗}} is equivalent to a geodesic distance. Hence (LHK) holds for any x, y ∈ K. □

3.4. Sierpinski carpets

In this section, we discuss another class of self-similar sets, the generalized Sierpinski carpets. The following definition is given by Barlow-Bass[7].

Definition 3.4.1. Let H_{0} = [0, 1]^{n}, where n ∈ ℕ, and let l ∈ ℕ with l ≥ 2. Set Q = {∏_{i=1}^{n}[(k_{i} − 1)/l, k_{i}/l] | (k_{1}, …, k_{n}) ∈ {1, …, l}^{n}}. For any Q ∈ Q, define F_{Q} : H_{0} → H_{0} by F_{Q}(x) = x/l + a_{Q}, where we choose a_{Q} so that F_{Q}(H_{0}) = Q. Let S ⊆ Q and let GSC(n, l, S) be the self-similar set with respect to {F_{Q}}_{Q∈S}, i.e. GSC(n, l, S) is the unique nonempty compact set satisfying GSC(n, l, S) = ∪_{Q∈S}F_{Q}(GSC(n, l, S)). Set H_{1}(S) = ∪_{Q∈S}F_{Q}(H_{0}). GSC(n, l, S) is called a generalized Sierpinski carpet if and only if the following four conditions (GSC1), …, (GSC4) are satisfied: (GSC1) (Symmetry) H_{1}(S) is preserved be all the isometries of the unit cube H_{0}. (GSC2) (Connected) H_{1}(S) is connected. (GSC3) (Non-diagonality) For any x ∈ H_{1}(S), there exists r_{0} > 0 such that int(H_{1}(S) ∩ B_{r}(x)) is nonempty and connected for any r ∈ (0, r_{0}), where B_{r}(x) = {y|y ∈ ℝ^{n}, |x − y| < r}. (GSC4) (Border included) The line segment between 0 and (1, 0, …, 0) is contained in H_{1}(S).

The Sierpinski carpet (Example 1.7.4) is equal to GSC(2, 3, S), where S = Q − {[1/3, 2/3]^{2}}. Also [0, 1]^{n} = GSC(n, l, Q) for any l ≥ 2.

In the rest of this section, we fix a generalized Sierpinski carpet GSC(n, l, S) and write K = GSC(n, l, S). Also L is the self-similar structure associated with K, i.e. L = (GSC(n, l, S), S, {F_{Q}}_{Q∈S}). Let ν be a self-similar measure with weight (1/N, …, 1/N), where N = #(S).

Definition 3.4.2. For k ∈ {1, …, n} and s ∈ [0, 1], define S_{k,s} = {Q|Q ∈ S, Q ∩ Φ_{k,s}}, where Φ_{k,s} is a hyperplane in ℝ^{n} defined by

Φ_{k,s} = {(x_{1}, …, x_{n})|x_{k} = s}.

Also let Ψ_{k,l} be the parallel translation in k-direction by 1/l; Ψ_{k,l}(x_{1}, …, x_{n}) = (y_{1}, …, y_{n}), where y_{i} = x_{i} if i ≠ k and y_{k} = x_{k} + 1/l. For Q_{1}, Q_{2} ∈ S, Q_{1} and Q_{2} are called k-neighbors if and only if Ψ_{k,l}(Q_{1}) = Q_{2} or Ψ_{k,l}(Q_{2}) = Q_{1}.

Let rf_{k} : ℝ^{n} → ℝ^{n} be the reflection in the hyperplane Φ_{k,1/2}. The symmetry condition (GSC1) ensures that rf_{k}(Q) ∈ S_{k,1} for any Q ∈ S_{k,1}. In this sense, we regard rf_{k} as a map from S_{k,0} to S_{k,1}. Note that rf_{k} is a bijection between S_{k,0} and S_{k,1}.

Proposition 3.4.3. The self-similar structure L associated with a generalized Sierpinski carpet is rationally ramified with a relation set

R_{∗} = {(S_{k,0}, S_{k,1}, rf_{k}, Q_{1}, Q_{2}) | k ∈ {1, …, n},

Q_{1}, Q_{2} ∈ S and they are k-neighbors.}

Combining the above proposition with Theorem 1.6.1, we obtain the following fact.

Proposition 3.4.4. A self-similar scale 𝒮(a) is locally finite with respect to L. if and only if a_{rf_{k}(Q)} = a_{Q} for any k = 1, …, n and any Q ∈ S_{k,0}.

In the series of papers [2, 3, 4, 5, 6, 7], Barlow and Bass have constructed a diffusion process on a generalized Sierpinski carpet and studied it extensively. For example, they have obtained elliptic and parabolic Harnack inequalities, Poincaré inequality and sub-Gaussian heat kernel estimate. Unfortunately, the Dirichlet form on L^{2}(K, ν) associated with their diffusion process is not necessarily self-similar. On the other hand, in [34], Kusuoka and Zhou have given a prescription of construction a self-similar Dirichlet form on a generalized Sierpinski carpet.

Combining the methods and results in [7] and [34] as in [22], we obtain a local regular Dirichlet form (E, F) on L^{2}(K, ν) which has the self-similarity in the following sense: for any u ∈ F and any Q ∈ S, u∘F_{Q} ∈ F and there exists r > 0 such that

E(u, u) = \frac{1}{r} ∑_{Q∈S} E(u∘F_{Q}, u∘F_{Q})

for any u ∈ Q. In fact, from Kusuoka-Zhou’s method, we have (SSF). Moreover, the corresponding diffusion process enjoys the same inequalities and estimates as the original one studied by Barlow and Bass. See [7, Remark 5.11] and the discussion after it. In particular, the associated heat kernel satisfies UHK and LHK for any x, y ∈ K, where β > 2, μ = ν and a distance d is the Euclidean distance. Note that ν(B_{r}(x, d)) = cr^{n} for any r > 0.

Barlow-Kumagai have studied a time change of this process in [8]. Let μ be a self-similar measure on L with weight (μ_{i})_{i∈S}. Define

F_{μ} = {u|u ∈ L^{2}(K, μ), there exists f ∈ F_{e} such that u = f for μ-a.e. x ∈ K}

and set E_{μ}(u, u) = E(H̃_{A}f, H̃_{A}f) for u ∈ F_{μ}, where f ∈ F_{e} and u = f for μ-a.e. x ∈ K, A is the quasi support of μ and (H̃_{A}u)(x) = E_{x}(u(X_{h_{A}})). (See [15, Section 6.2] for details on time changes of a diffusion process associated with a Dirichlet form in general. Also see [8, p. 9].) In [8], they have shown that if μ_{Q}r < 1 for any Q ∈ S, then (E_{μ}, F_{μ}) is a local regular Dirichlet form on L^{2}(K, μ) and the associated diffusion process is a time change of the diffusion associated with (E, F). By their discussion, we can verify (SSF), (PI), (CHK) and (UPH).

Here after, we fix a self-similar measure μ with weight (μ_{Q})_{Q∈S} and assume that μ_{Q}r < 1 for any Q ∈ S. The following lemma is immediate by Theorems 1.3.5, 1.6.6 and Proposition 3.4.4.

Theorem 3.4.5. Define 𝒮_{∗} = 𝒮(γ), where γ_{Q} = \sqrt{μ_{Q}r} for any Q ∈ S and γ = (γ_{Q})_{Q∈S}. Then μ has the volume doubling property with respect to 𝒮_{∗} if and only if μ_{Q} = μ_{rf_{k}(Q)} for any k = 1, …, n and any Q ∈ S_{k,0}.

a  b  a c  c a  b  a  ^{a = μ_{1} = μ_{3} = μ_{5} = μ_{7}}^{b = μ_{2} = μ_{6}}^{c = μ_{4} = μ_{8}}

Figure 3.3. Self-similar volume doubling measures on the Sierpinski carpet

This theorem shows when the condition (a) of Theorem 3.2.3 holds. Consequently, the claims of Theorem 3.2.3 follows if μ_{Q} = μ_{rf_{k}(Q)} for any k = 1, …, n and any Q ∈ S_{k,0}. In particular, if D_{γ^{α}} is a distance, set β = 2/α and d(x, y) = D_{γ^{α}}(x, y)^{1/α}. Then

\frac{c_{1}}{μ(B\sqrt{t}(x, d))} ≤ p(t, x, x) ≤ \frac{c_{2}}{μ(B\sqrt{t}(x, d))}

and

p(t, x, y) ≤ \frac{c_{3}}{μ(B\sqrt{t}(x, d))} exp ( − c_{4}(\frac{d(x, y)^{2}}{t} )^{\frac{1}{β−1}} )

for any t ∈ (0, 1] and any x, y ∈ K, where p(t, x, y) is the heat kernel associated with the Dirichlet form (E_{μ}, F_{μ}) on L^{2}(K, μ). Moreover, we have the elliptic Harnack inequality by [7]. (Note that harmonic functions associated with (E_{μ}, F_{μ}) on L^{2}(K, μ) are the same as those associated with (E, F) on L^{2}(K, ν).) Also we have the exit time estimate (E) by Lemma 3.5.13. Using the arguments in [17], we have the near diagonal lower estimate (3.5.8). Hence, if D_{γ^{α}} is equivalent to a geodesic distance, then the classical arguments in [1, 8, 18, 30] imply the lower off-diagonal Li-Yau estimate (LHK).

Finally we present two examples.

Example 3.4.6 (the Sierpinski carpet). Let L = (K, S, {F_{i}}_{i∈S}) be the selfsimilar structure associated with the Sierpinski carpet appearing in Examples 1.5.12 and 1.7.4. By [2, 3] and [34], the resistance scaling ratio r is less than one and hence we are in the recurrent case. By Theorem 3.4.5, the condition (a) of Theorem 3.2.3 follows if and only if μ_{1} = μ_{3} = μ_{5} = μ_{7}, μ_{2} = μ_{6} and μ_{4} = μ_{8}. See Figure 3.3. Furthermore, if μ_{2} = μ_{4} and μ_{1} ≤ μ_{4} as well, then

(0, α_{∗}] = {α|D_{γ^{α}} is a distance},

A  B  A B  B A  B  A  ^{A = (μ_{1}r)^{α_{∗}/2}}^{B = (μ_{2}r)^{α_{∗}/2}}^{A ≤ B} 2A + B = 1

Figure 3.4. Geodesic distances on the Sierpinski carpet

where α_{∗} is given by 2(μ_{1}r)^{α_{∗}/2} + (μ_{2}r)^{α_{∗}/2} = 1, and D_{γ^{α∗}} is equivalent to a geodesic distance. See Figure 3.4. Details on the construction of geodesic distances on the Sierpinski carpet can be found in [32]. In this case, we have the upper and lower off-diagonal Li-Yau estimates (UHK) and (LHK):

\frac{c_{1}}{μ(B\sqrt{t}(x, d))} exp ( − c_{2}(\frac{d(x, y)^{2}}{t} )^{1/(β_{∗}−1)}) ≤ p(t, x, y)

≤ \frac{c_{3}}{μ(B\sqrt{t}(x, d))} exp ( − c_{4}(\frac{d(x, y)^{2}}{t} )^{1/(β_{∗}−1)})

for any x, y ∈ K and any t ∈ (0, 1], where d(x, y) = (D_{γ^{α∗}})^{1/α_{∗}} and β_{∗} = 2/α_{∗}.

Example 3.4.7 (Cubes). Let l = 3 and let S = Q. Then K = [0, 1]^{n}. In this case, ν is the restriction of the Lebesgue measure,

F = H_{1}(K) = {f|f : K → ℝ, all the partial derivatives of f

in the sense of distribution belong to L^{2}(K, ν)}

and

E(u, v) = ∑_{k=1}^{n} ∫_{K} \frac{∂u}{∂x_{k}} \frac{∂v}{∂x_{k}} dν,

where ∂u/∂x_{i} is the derivative in the sense of distribution. The diffusion process associated with the Dirichlet form (E, F) on L^{2}(K, ν) is the reflected Brownian motion. In this case, for any u, v ∈ F,

E(u, v) = 3^{2−n} ∑_{Q∈S} E(u∘F_{Q}, v∘F_{Q}).

Hence r = 3^{n−2}. Hence we are not in the recurrent case unless n = 1. If μ_{Q} < 3^{2−n} for any Q ∈ S, then we have a local regular Dirichlet form (E_{μ}, F_{μ}) on L^{2}(K, μ), where μ is the self-similar measure with weight (μ_{Q})_{Q∈S}. The corresponding diffusion process is the time change of the reflected Brownian motion on n-dimensional cube [0, 1]^{n}. In particular, if n = 2, then r = 1 and (E_{μ}, F_{μ}) is a local regular Dirichlet form on L^{2}(K, μ) for any self-similar measure μ. Applying Theorem 3.4.5, we obtain Theorem 0.2.5.

3.5. Proof of Theorem 3.2.3

As in Section 3.2, (K, S, {F_{i}}_{i∈S}) is a rationally finite self-similar structure and (E, F) is a local regular Dirichlet form on L^{2}(K, μ) which is conservative and satisfies (SSF), (PI), (CHK) and (UPH). Also 𝒮_{∗} is the scale induced by the gauge function g(w) = \sqrt{r_{w}μ(K_{w})}. We write 𝒮_{∗} = \{Λ_{s}\}_{0<s≤1}.

First note that Theorem 2.3.17 implies the following equivalence.

Lemma 3.5.1. (a) is equivalent to (b).

Definition 3.5.2. Let U be a nonempty open subset of K. Define D_{U} = {u|u ∈ F ∩ C(K), u|_{K\U} ≡ 0} and

λ_{∗}(U) = inf_{u∈D_{U}} \frac{E(u, u)}{||u||_{2}^{2}}.

Also define F_{U} by the closure of D_{U} with respect to the inner product E_{∗}(u, v) = E(u, v) + ∫_{K} uvdμ.

Proposition 3.5.3. Let U be a nonempty open subset of K. If E_{U} = E_{F_{U} ×F_{U}}, then (E_{U}, F_{U}) is a local regular Dirichlet form on L^{2}(K, μ) (or L^{2}(U, μ|_{U})). If −Δ_{U} is the self-adjoint operator on L^{2}(K, μ) associated with (E_{U}, F_{U}), then −Δ_{U} has compact resolvent and λ_{∗}(U) is the minimal eigenvalue of −Δ_{U}. Also if p_{U}(t, x, y) is the heat kernel associated with the Dirichlet form (E_{U}, F_{U}), then. for any t > 0,

0 ≤ p_{U}(t, x, y) ≤ p(t, x, y)

for μ × μ-a.e. (x, y) ∈ K^{2}.

Lemma 3.5.4. There exists c > 0 such that, for any w ∈ W_{∗},

λ_{∗}(B_{w}) ≤ \frac{c}{r_{w}μ(K_{w})},

where B_{w} = K_{w}\F_{w}(\overline{V}_{0}).

Proof. Choose v ∈ W_{∗} so that K_{v} ⊆ K\\overline{V}_{0}. Since (E, F) is a regular Dirichlet form, there exists φ ∈ C(K) ∩ F such that supp(φ) ⊆ K\\overline{V}_{0} and φ(x) ≥ 1 for any x ∈ K_{v}. Define φ_{w} by

φ_{w}(x) = \begin{cases}φ((F_{w})^{−1}(x)) & if x ∈ K_{w}, \\ 0 & otherwise. \end{cases}

Then by (SSF), φ_{w} ∈ F_{B_{w}} and E(φ_{w}, φ_{w}) = (r_{w})^{−1}E(φ, φ). Since μ is elliptic, ||φ_{w}||_{2}^{2} ≥ μ(K_{wv}) ≥ c′μ(K_{w}), where c′ is independent of w. Therefore,

λ_{∗}(B_{w}) ≤ \frac{E(φ_{w}, φ_{w})}{||φ_{w}||_{2}^{2}} ≤ \frac{E(φ, φ)}{c′r_{w}μ(k_{w})}.

□

Lemma 3.5.5. (d) implies (b).

Proof. Choose α > 0 so that d^{α} is a distance. Let D(x, y) = d(x, y)^{α} and let β = 2/α. Then by (d),

p(t, x, x) ≤ \frac{c_{1}}{μ(B_{t^{1/β}}(x, D))}.

Since d is adapted to the scale 𝒮_{∗}, there exists c_{2} > 0 such that U_{c_{2}s}(x) ⊆ B_{s}(x, d) for any x ∈ K and any s ∈ (0, 1]. Hence for any r > 0, U_{c_{2}s}(x) ⊆ B_{s}(x, d) = B_{r}(x, D), where s = r^{1/α}. Let w ∈ Λ_{c_{2}s,x}. Then by Lemma 3.5.4,

λ_{∗}(B(r, D)) ≤ λ_{∗}(B_{w}) ≤ \frac{c_{3}}{r_{w}μ(K_{w})} ≤ c_{4}r^{−β}.

Using Theorem C.3, we have the volume doubling property of μ with respect to the distance D. This immediately implies (b). □

Lemma 3.5.6. If there exist positive constants c_{1} and c_{2} such that

p(t, x, x) ≤ \frac{c_{1}}{μ(U_{c_{2}\sqrt{t}}(x))}

for any x ∈ X and any t ∈ (0, 1], then μ has the volume doubling property with respect to 𝒮_{∗}. In particular, (c) implies (a).

Remark. In the following proof, we don’t need the assumption (I) neither (II).

Proof. Let s = c_{2}\sqrt{t} and let x ∈ K. If w ∈ Λ_{s,x}, then U_{s}(y) = U_{s}(w) ⊆ U_{s}(x) for any y ∈ B_{w}, where U_{s}(w) = K(W (Λ_{s}, K_{w})). By (c) and Proposition 3.5.3,

p_{B_{w}}(t, y, y) ≤ p(t, y, y) ≤ \frac{c_{1}}{μ(U_{s}(y))}.

Integrating this over B_{w}, we see that

e^{−λ_{∗}(B_{w})t} ≤ ∫_{B_{w}} p_{B_{w}}(t, y, y)μ(dy) ≤ \frac{c_{1}μ(K_{w})}{μ(U_{s}(y))}.

By Lemma 3.5.4, it follows that c_{∗} ≤ e^{−λ_{∗}(B_{w})t}, where c_{∗} is independent of x, t and w ∈ Λ_{s,x}. Hence,

c_{∗}μ(U_{s}(w)) ≤ c_{1}μ(K_{w})

for any w ∈ Λ_{s,x}. Since ∪_{w∈Λ_{s,x}}U_{s}(w) = U_{s}(x),

c_{∗}μ(U_{s}(x)) ≤ c_{∗} ∑_{w∈Λ_{s,x}} μ(U_{s}(w)) ≤ c_{1} ∑_{w∈Λ_{s,x}} μ(K_{w}) = c_{1}μ(K_{s}(x)).

This is the condition (A)_{1} in Section 1.3. Since both μ and 𝒮_{∗} are elliptic, Theorem 1.3.10 implies the condition (VD)_{0}. Hence by Theorem 1.3.5, we have (a). □

Lemma 3.5.7. (d) implies (c).

Proof. Note that by the previous lemmas, we have (b) and (a). Since d is adapted to the scale 𝒮_{∗}, there exist n ≥ 1 and c_{1} > 0 such that U_{c_{1}s}^{(n)}(x) ⊆ B_{s}(x, d) for any x ∈ X and any s ∈ (0, 1]. Therefore by (d),

p(t, x, x) ≤ \frac{c}{μ(U_{c_{1}\sqrt{t}}^{(n)}(x))} ≤ \frac{c}{μ(U_{c_{1}\sqrt{t}}(x))}.

Now the volume doubling property of μ with respect to the scale 𝒮_{∗} immediately implies (c). □

Lemma 3.5.8. If (a) is satisfied, then there exists c > 0 such that r_{w} ≤ cr_{v} for any s ∈ (0, 1] and any w, v ∈ Λ_{s} with K_{w} ∩ K_{v} ≠ ∅.

Proof. By Theorem 1.3.5, μ is gentle with respect to 𝒮_{∗}. Hence there exists c_{1} > 0 such that μ(K_{w}) ≤ cμ(K_{v}) and r_{w}μ(K_{w}) ≤ c_{1}r_{v}μ(K_{v}) for any s ∈ (0, 1] and any w, v ∈ Λ_{s} with K_{w} ∩ K_{v} ≠ ∅.This shows that r_{w} ≤ (c_{1})^{2}r_{v}. □

Proposition 3.5.9. Assume that (E, F) is recurrent. For any closed subset B of K, there exists g_{B} : K × K → [0, ∞) which has the following properties: (GF1) g_{B}(x, y) = g_{B}(y, x) ≤ g_{B}(x, x) for any x, y ∈ K.For any x ∈ K, define g_{B}^{x} by g_{B}^{x}(y) = g_{B}(x, y). Then g_{B}^{x} ∈ F_{X\B} and E(g_{B}^{x}, u) = u(x) for any x ∈ K and any u ∈ F_{X\B}. (GF2) |g_{B}(x, y) − g_{B}(x, z)| ≤ R(y, z) for any y, z ∈ K. (GF3) For x ∉ B, define

R(x, B) = ( min{E(u, u)|u ∈ F_{X\B}, u(x) = 1})^{−1}.

Then g_{B}(x, x) = R(x, B) > 0. (GF4) For x ∉ B,

E_{x}(h_{B}) = ∫_{K} g_{B}(x, y)μ(dy).

g_{B} is called the B-Green function.

Proof. Since (E, F) is recurrent, (E, F) is a resistance form on K. If B is a finite set, the above results are shown in [29]. Generalization to a closed set is straight forward. See [27] for details. □

Lemma 3.5.10. Assume (a). Set V_{s}^{(n)}(x) = int(U_{s}^{(n)}(x)) for any (s, x) ∈ (0, 1]× K and define E_{s,x}(·, ·) by

E_{s,x}(u, u) = ∑_{v∈Λ_{s,x}^{n}} E(u ∘ F_{v}, u ∘ F_{v})

for u ∈ F_{V_{s}^{(n)}(x)}.Then there exist c_{1}, c_{2} > 0 such that

\frac{c_{1}}{r_{w}} E_{s,x}(u, u) ≤ E_{V_{s}^{(n)}(x)}(u, u) ≤ \frac{c_{2}}{r_{w}} E_{s,x}(u, u)

for any (s, x) ∈ (0, 1] × K and any u ∈ F_{V_{s}^{(n)}(x)}.

Proof. By (SSH), if u ∈ F_{V_{s}^{(n)}(x)},

E_{V_{s}^{(n)}(x)}(u, u) = ∑_{v∈Λ_{s,x}^{n}} \frac{1}{r_{v}} E(u ∘ F_{v}, u ∘ F_{v})

Using Lemma 3.5.8, we immediately deduce the desired inequality. □

Lemma 3.5.11. Assume (a) and that (E, F) is recurrent. For (s, x) ∈ (0, 1]×K, define

\overline{R}_{s,x} = sup_{y∈K_{s}(x)} R_{s,x}(y) and \underline{R}_{s,x} = inf_{y∈K_{s}(x)} R_{s,x}(y),

where

R_{s,x}(y) = ( inf{E_{s,x}(u, u)|u ∈ F_{V_{s}^{(n)}(x)}, u(y) = 1})^{−1}.

Then, 0 < \underline{R}_{s,x} ≤ \overline{R}_{s,x} < +∞ and

(c_{2})^{−1}r_{w}\underline{R}_{s,x} ≤ R(x, V_{s}^{(n)}(x)^{c}) ≤ (c_{1})^{−1}r_{w}\overline{R}_{s,x}

for any w ∈ Λ_{s,x}, where c_{1} and c_{2} are the same constants as in Lemma 3.5.10

Proof. By Lemma 3.5.10, for any w ∈ Λ_{s,x},

(3.5.1)  (c_{2})^{−1}r_{w}R_{s,x}(y) ≤ R(y, V_{s}^{(n)}(x)^{c}) ≤ (c_{1})^{−1}r_{w}R_{s,x}(y).

Since R(y, V_{s}^{(n)}(x)^{c}) = g_{V_{s}^{(n)}(x)^{c}}(y, y), it follows that

c_{1}g_{V_{s}^{(n)}(x)^{c}}(y, y) ≤ r_{w}R_{s,x}(y) ≤ c_{2}g_{V_{s}^{(n)}(x)^{c}}(y, y).

Note that g_{V_{s}^{(n)}(x)^{c}}(y, y) is continuous with respect to y and is positive for any

y ∈ K_{s}(x) ⊆ V_{s}^{(n)}(x). Therefore we see that 0 < \underline{R}_{s,x} ≤ \overline{R}_{s,x} < +∞ because K_{s}(x) is compact. Now the desired result is straight forward from (3.5.1). □

Lemma 3.5.12. Assume that (E, F) is recurrent. If (a) holds, then there exist positive constants c_{3} and c_{4} such that

(RES)  c_{3}r_{w} ≤ R(x, V_{s}^{(n)}(x)^{c}) ≤ c_{4}r_{w}

for any x ∈ K, any s ∈ (0, 1] and any w ∈ Λ_{s,x}.

Proof. Suppose that (s, x) ∼_{n+1} (t, y). Let ψ be the n+1-isomorphism between

(s, x) and (t, y) and let ϕ be the associated L-similitude between U_{s}^{(n+1)}(x) and U_{t}^{(n+1)}(y). Note that ψ(Λ_{s.x}^{k}) = Λ_{t,y}^{k} for k = 0, 1, …, n + 1, ϕ(U_{s}^{(n)}(x)) = U_{s}^{(n)}(y) and ϕ(K_{s}(x)) = K_{s}(y). Since ϕ(∂U_{s}^{(n)}(x)) = ∂U_{t}^{(n)}(y), it follows from (SSH) that ϕ_{∗} : F_{V_{t}^{(n)}(y)} → F_{V_{s}^{(n)}(x)} defined by ϕ_{∗}(u) = u∘ϕ is bijective. Moreover,

E_{s,x}(ϕ_{∗}(u), ϕ_{∗}(u)) = ∑_{v∈Λ_{s,x}^{n}} E(ϕ_{∗}(u) ∘ F_{v}, ϕ_{∗}(u) ∘ F_{v})

= ∑_{v∈Λ_{s,x}^{n}} E(u ∘ F_{ψ(v)}, u ∘ F_{ψ(v)}) = E_{t,y}(u, u).

Hence R_{s,x}(z) = R_{t,y}(ϕ(z)). So \overline{R}_{s,x} and \underline{R}_{s,x} depend only on the equivalence classes under ∼_{n+1}. By Theorem 1.3.5, (a) implies that 𝒮_{∗} is locally finite. Hence by

Theorems 2.2.7 and 2.2.13, the number of equivalence classes under ∼_{n+1} is finite.

Now Lemma 3.5.11 suffices to deduce the lemma. □

Lemma 3.5.13. Assume that (E, F) is recurrent. If (a) holds, then there exists c_{5}, c_{6} > 0 such that

(E)  c_{5}s^{2} ≤ E_{x}(h_{V_{s}^{(n)}(x)^{c}}) ≤ c_{6}s^{2}

for any (s, x) ∈ (0, 1] × K.

Proof. First we show the upper estimate. By Proposition 3.5.9,

E_{x}(h_{V_{s}^{(n)}(x)^{c}}) = ∫_{V_{s}^{(n)}(x)} g_{V_{s}^{(n)}(x)^{c}}(x, y)μ(dy) ≤ R(x, V_{s}^{(n)}(x)^{c})μ(V_{s}^{(n)}(x)).

Since μ is gentle with respect to 𝒮_{∗} and 𝒮_{∗} is locally finite, μ(V_{s}^{(n)}(x)) ≤ cμ_{w} for w ∈ Λ_{s,x}, where c is independent of s, x and w.This along with Lemma 3.5.12 yields

E_{x}(h_{V_{s}^{(n)}(x)^{c}}) ≤ cc_{4}r_{w}μ_{w} ≤ c_{6}s^{2}.

For the lower estimate, note that (SSH) implies

(3.5.2)  R(F_{v}(y), F_{v}(z)) ≤ r_{v}R(y, z)

for any y, z ∈ K and any v ∈ W_{∗}. (See [28, Lemma 3.3.5] for details.) Hence sup_{y,z∈K_{v}} R(y, z) ≤ Mr_{v} for any v ∈ W_{∗}, where M = sup_{p,q∈K} R(p, q). Choose m so that M(max_{i∈S} r_{i})^{m} ≤ c_{3}/2. Then, for any w ∈ Λ_{s,x}, there exists v ∈ W_{m} such that x ∈ K_{wv} and R(x, y) ≤ c_{3}r_{w}/2 for any y ∈ K_{wv}. By (GF2) and Lemma 3.5.12,

g_{V_{s}^{(n)}(x)^{c}}(x, y) ≥ g_{V_{s}^{(n)}(x)^{c}}(x, x) − R(x, y) ≥ c_{3}r_{w}/2

for any y ∈ K_{wv}. Therefore,

E_{x}(h_{V_{s}^{(n)}(x)^{c}}) ≥ ∫_{K_{wv}} g_{V_{s}^{(n)}(x)^{c}}(x, y)μ(dy) ≥ c_{3}r_{w}μ(K_{wv})/2.

Since μ is elliptic, μ(K_{wv}) ≥ bμ(K_{w}), where b is independent of s, x and w. Therefore we obtain the lower estimate. □

Lemma 3.5.14. Assume (a). For any (s, x) ∈ (0, 1] × K, define μ_{s,x} by

μ_{s,x} = ∑_{v∈Λ_{s,x}^{n}} μ(F_{w}^{−1}(A ∩ K_{v}))

for any Borel set A ⊆ K. If μ is self-similar, then there exist c_{5}, c_{6} > 0 such that

c_{5}μ_{w}μ_{s,x}(A) ≤ μ(A) ≤ c_{6}μ_{w}μ_{s,x}(A)

for any (s, x) ∈ (0, 1] × K, any w ∈ Λ_{s,x} and any Borel set A ⊆ U_{s}(x).

Lemma 3.5.15. For (s, x, w) ∈ (0, 1] × K × Λ_{s,x}, let {E_{y}^{s,x,w}(·)}_{y∈V_{s}^{(n)}(x)} be the expectation with respect to the diffusion process associated with the local regular Dirichlet form (r_{w}E_{V_{s}^{(n)}(x)}, F_{V_{s}^{(n)}(x)}) on L^{2}(K, μ_{s,x}). Define

\underline{E}^{s,x,w} = inf_{y∈K_{s}(x)} E_{y}^{s,x,w}(h_{V_{s}^{(n)}(x)^{c}}) and \overline{E}^{s,x,w} = sup_{y∈K_{s}(x)} E_{y}^{s,x,w}(h_{V_{s}^{(n)}(x)^{c}}).

If (a) is satisfied, then 0 < \underline{E}^{s,x,w} ≤ \overline{E}^{s,x,w} < +∞ and

c_{7}\underline{E}^{s,x,w}s^{2} ≤ E_{x}(h_{V_{s}^{(n)}(x)^{c}}) ≤ c_{8}\overline{E}^{s,x,w}s^{2},

where c_{7} and c_{8} are independents of (s, x, w).

Proof. Use h(t, y, z) to denote the heat kernel associated with the Dirichlet form (r_{w}E_{V_{s}^{(n)}(x)}, F_{V_{s}^{(n)}(x)}) on L^{2}(K, μ_{s,x}). Recall that p_{V_{s}(x)}(t, y, z) is the heat kernel associated with the Dirichlet form (E_{V_{s}^{(n)}(x)}, F_{V_{s}^{(n)}(x)}) on L^{2}(K, μ). Therefore, by Lemma 3.5.14,

∫_{V_{s}^{(n)}(x)} h(\frac{t}{c_{5}μ_{w}r_{w}} , y, z)μ_{s,x}(dz) ≤ ∫_{V_{s}^{(n)}(x)} p_{V_{s}^{(n)}(x)}(t, y, z)μ(dz)

≤ ∫_{V_{s}^{(n)}(x)} h(\frac{t}{c_{6}μ_{w}r_{w}} , y, z)μ_{s,x}(dz),

where c_{5} and c_{6} are the same constants as in Lemma 3.5.14. Integrating this on [0, ∞) with respect to t, we obtain

(3.5.3)  c_{5}μ_{w}r_{w}E_{y}^{s,x,w}(h_{B}) ≤ E_{y}(h_{B}) ≤ c_{6}μ_{w}r_{w}E_{y}^{s,x,w}(h_{B}),

where B = V_{s}^{(n)}(x)^{c}. Note that h(t, y, z) has uniform exponential decay for sufficiently large t, i.e. there exist c, λ > 0 and t_{∗} > 0 such that h(t, y, z) ≤ ce^{−λt} for any y, z and t ≥ t_{∗}. Hence \overline{E}^{s,x,w} < +∞. By (UPH), ∫_{y∈K_{s}(x)} E_{y}(h_{B}) > 0. Hence (3.5.3) implies that \underline{E}^{s,x,w} > 0. Again using (3.5.3), we obtain the desired inequality. □

Lemma 3.5.16. For any (s, x), (t, y) ∈ (0, 1] × K, we write (s, x) ∼_{∗} (t, y) if and only if (s, x) ∼_{n+1} (t, y) and there exists c > 0 such that r_{ψ(w)} = cr_{w} for any

w ∈ Λ_{s,x}^{n}, where ψ : Λ_{s,x}^{n+1} → Λ_{t,y}^{n+1} is the n + 1-isomorphism between (s, x) and (t, y). Assume (a). If (II) is satisfied, i.e. μ is self-similar and the resistance scaling ratio is arithmetic on R_{1}-relations, then (0, 1] × K/∼_{∗} is finite.

Proof. By Theorem 1.3.5, 𝒮_{∗} is locally finite and 𝒮_{∗} ∼_{GE} 𝒮_{μ}, where 𝒮_{μ} is the

scale induced by μ. Hence by Theorem 1.4.3-(1), 𝒮_{μ} is locally finite as well. Let (X, Y, φ, x, y) ∈ R_{2}, where R is the relation set of L. If (X, Y, φ, x, y) ∈ R_{2}, then Theorem 1.6.1 yields that r_{w}μ_{w} = r_{φ(w)}μ_{φ(w)} and μ_{φ(w)} = μ_{φ(w)} for any w ∈ X. Hence r_{w} = r_{φ(w)} for any w ∈ X.

Next we show that {r_{w}/r_{v}|(w, v) ∈ IP(L, 𝒮)} is a finite set. At first, let (w, v) ∈ IP(L, 𝒮, R_{2}). Then the above discussion along with Lemma 3.5.8 implies that the choice of the values r_{w}/r_{v} is finite.If (w, v) ∈ IP(L, 𝒮, R_{1}, then we also have finite number of choices of r_{w}/r_{v}, because (r_{i})_{i∈S} is arithmetic on R_{1}-relation. Hence {r_{w}/r_{v}|(w, v) ∈ IP(L, 𝒮, R)} is a finite set. Now let (w, v) ∈ IP(L, 𝒮). As in the proof of Theorem 2.2.7, we have {w(i)}_{i=1,...,m+1} which satisfies w(1) = w, w(m + 1) = v and (w(i), w(i + 1)) ∈ IP(L, 𝒮, R) for any i. Note that m + 1 ≤ inf_{p∈K} #(π^{−1}(p)) < ∞. This fact along with the finiteness of {r_{w}/r_{v}|(w, v) ∈ IP(L, 𝒮, R)} implies that {r_{w}/r_{v}|(w, v) ∈ IP(L, 𝒮)} is a finite set.

Now by Theorems 2.2.7 and 2.2.13, the number of equivalence classes under ∼_{n+1} is finite. Since we only have finite number of choices of r_{w}/r_{v} for (w, v) ∈

IP(L, 𝒮_{∗}), one equivalent class of ∼_{n+1} contains finite number of equivalent classes

of ∼_{∗}. Therefore, (0, 1]/∼_{∗} is a finite set. □

Lemma 3.5.17. Under the assumption (II), (a) implies (E).

Proof. Let (s, x) ∼_{∗} (t, y), let ψ : Λ_{s,x}^{n+1} → Λ_{t,y}^{n+1} be the associated n + 1-

isomorphism and let ϕ : U_{s}^{(n+1)}(x) → U_{t}^{(n+1)}(y) be the associated similitude. Choose w ∈ Λ_{s,x}. Then ϕ gives a natural correspondence between the Dirichlet forms (r_{w}E_{V_{s}^{(n)}(x)}) on L^{2}(V_{s}^{(n)}(x), μ_{s,x}) and (r_{ψ(w)}E_{V_{t}^{(n)}(y)}) on L^{2}(V_{t}^{(n)}(y), μ_{t,y}).

Therefore, \underline{E}^{s,x,w} = \underline{E}^{t,y,ψ(w)} and \overline{E}^{s,x,w} = \overline{E}^{t,y,ψ(w)}. Hence Lemmas 3.5.15 combined with 3.5.16 suffices for (E). □

Lemma 3.5.18. Assume (b). Then (DLHK) and (UHK) holds with β > 1. In particular, (b) implies (d).

Proof. Note that we have (a) as well due to Theorem 2.3.17. Since d is adapted to 𝒮_{∗}, U_{cr}^{(n)}(x) ⊆ B_{r}(x, d) ⊆ U_{c′r}^{(n)}(x). Hence by Lemmas 3.5.13 and 3.5.17, c_{5}c^{2}r^{2} ≤ E_{x}(h_{B_{r}(s,d)^{c}}) ≤ c_{6}c′^{2}r^{2}. Let D(·, ·) = d(·, ·)^{α}.Recall that β = 2/α. Then, we have the exit time estimate with respect to the distance D:

(3.5.4)  a_{1}r^{β} ≤ E_{x}(h_{B_{r}(x,D)^{c}}) ≤ a_{2}r^{β}.

Since μ is gentle with respect to 𝒮_{∗} and 𝒮_{∗} is locally finite, there exists γ > 0 such that

(3.5.5)  γμ(U_{s}^{(n)}(x)) ≤ μ(K_{w}) ≤ μ(U_{s}^{(n)}(x))

for any (s, x) ∈ (0, 1] × K and any w ∈ Λ_{s,x}. Recall that Λ_{s}(u) = {w|w ∈ Λ_{s}, K_{w} ∩ supp(u) ≠ ∅} for u ∈ F. By (3.5.5),

γ min_{x∈supp(u)} μ(U_{s}^{(n)}(x)) ≤ min_{w∈Λ_{s}(u)} μ(K_{w}) ≤ min_{x∈supp(u)} μ(U_{s}^{(n)}(x)).

Combining this with (3.1.2), we obtain the local Nash inequality:

E(u, u) + \frac{a}{r^{β} inf_{x∈supp(u)} μ(B_{r}(x, D))}||u||_{1}^{2} ≥ \frac{b}{r^{β}} ||u||_{2}^{2}

for any u ∈ F and any r ∈ (0, 1], where a and b are independent of u.

Now using [30, Theorem 2.9 and Theorem 2.13], we see that β > 1 and obtain (DLHK) and (UHK). □

By the above lemmas, we see that (a), (b), (c) and (d) are all equivalent. Now the remaining part of a proof is to show (LHK) under given assumptions.

Lemma 3.5.19. Assume (a). For any ϵ > 0, there exists γ > 0 such that R(x, y)μ(U_{s}^{(n)}(x)) ≤ ϵs^{2} for any x ∈ K, any y ∈ U_{γs}^{(n)}(x) and any s ∈ (0, 1].

Proof. Write V (s, x) = μ(U_{s}^{(n)}(x)). By (a), 𝒮_{∗} is locally finite and μ is gentle with respect to 𝒮_{∗}. Let (s, x) ∈ (0, 1]×K and let w ∈ Λ_{s,x}. Then V (s, x) ≤ cμ(K_{w}) and s^{2} ≥ cr_{w}μ(K_{w}), where c is independent of (s, x, w). Hence,

(3.5.6)  \frac{R(x, y)V (s, x)}{s^{2}} ≤ c′ \frac{R(x, y)}{r_{w}}.

Now since 𝒮_{∗} is elliptic, for any m ≥ 1, we can choose γ ∈ (0, 1) so that |v| ≥ m if w′ ∈ Λ_{s} and w′v ∈ Λ_{γs}. For any y ∈ U_{γs}^{(n)}(x), there exists {w(i)v(i)}_{i=0}^{n} ∈ Λ_{γs,x}^{n} such that w(0) = w, w(k) ∈ Λ_{s,x}^{n} and K_{w(k−1)v(k−1)} ∩ K_{w(k)v(k)} ≠ ∅ for any k = 1, …, n. Since r_{w′} ≤ ar_{w} for any w′ ∈ U_{s}^{(n)}(x), where a is independent of s, x and w, (3.5.2) shows that

R(x, y) ≤ R_{∗} ∑_{k=0}^{n} r_{w(k)}r_{v(k)} ≤ a(n + 1)Mr_{w}(r_{∗})^{m},

where M = sup_{p,q∈K} R(p, q) and r_{∗} = max_{i∈S} r_{i}. Choosing a sufficiently large m, we verify the statement of the lemma from (3.5.6). □

Lemma 3.5.20. Assume (a). In the recurrent case, (LHK) holds for any geodesic pair for d^{α}.

Proof. Let p^{t,x}(y) = p(t, x, y) for any t, x, y. Then p^{t,x} belongs to the domain of the self-adjoint operator associated with the Dirichlet from (E, F) on L^{2}(K, μ). By the definition (RF4),

(3.5.7) |p(t, x, y) − p(t, x, x)|^{2} ≤ E(p^{t,x}, p^{t,x})R(x, y)

≤ −\frac{∂p}{∂t}(2t, x, x)R(x, y) ≤ \frac{p(t, x, x)R(x, y)}{t}.

(This inequality has been obtained in [14, Lemma 6.4] and [21, Lemma 5.2].) Combining (3.5.7) with (DLHK) and (c), we obtain

p(t, x, y) ≥ p(t, x, x)(1 − \sqrt{\frac{R(x, y)}{tp(t, x, x)}})

≥ \frac{c}{μ(B\sqrt{t}(x, d))}(1 − c′\sqrt{\frac{R(x, y)μ(U_{s}^{(n)}(x))}{t}})

By Lemma 3.5.19, there exists γ > 0 such that R(x, y)μ(U_{√\overline{t}}^{(n)}(x))/t ≤ (c′/2)^{2} for

any y ∈ U_{γ\sqrt{t}}^{(n)}(x). Since d is adapted to 𝒮_{∗},

p(t, x, y) ≥ \frac{c′′}{μ(B\sqrt{t}(x, d))}

for any y ∈ B_{δ\sqrt{t}}(x, d). Let D = s^{α}. Rewriting this in terms of D, we have

(3.5.8)  p(t, x, y) ≥ \frac{c′′′}{μ(B_{t^{1/β}}(x, D))}

for any y ∈ B_{δ′t^{1/β}}(x, D). This is so called the near diagonal lower estimate. Note that we also have the exit time estimate (3.5.4) and the volume doubling property. By the argument of the proof of [30, Theorem 2.13], we obtain (LHK) for geodesic pairs. □

Remark. In [30, Theorem 2.13], it is assumed that the distance is a geodesic distance. However, the discussion of the proof of [30, Theorem 2.13] can get through if there exists a geodesic between given two points. The constants are determined by those appeared in the near diagonal estimate and the volume doubling property, and hence they do not depends on the points.

Appendix

A. Existence and continuity of a heat kernel

Let (X, d) be a locally compact metric space and let μ be a Radon measure on (X, d). Let (E, F) be a regular Dirichlet form on L^{2}(X, μ). We use H to denote the non-negative self-adjoint operator from L^{2}(X, μ) to itself. Also let {T_{t}}_{t>0} be the strongly continuous semigroup associated with H, i.e.T_{t} = e^{−tH}.

Definition A.1. The semigroup {T_{t}}_{t>0} is said to be ultracontractive if and only if T_{t} can be extended to a bounded operator from L^{2}(X, μ) to L^{∞}(X, μ) for any t > 0.

Note that T_{t} is self-adjoint. Using the duality, T_{t} can be extended to a bounded operator from L^{1}(X, μ) to L^{∞}(X, μ) as well if {T_{t}}_{t>0} is ultracontractive.

One of the conditions implying the ultracontractivity is the Nash inequality.

Notation. ||·||_{p} is the L^{p}-norm of L^{p}(X, μ) Also.||A||_{p→q} is the operator norm of a bounded linear operator A : L^{p}(X, μ) → L^{q}(X, μ).

Theorem A.2. For α > 0, the following conditions (1), (2) and (3) are equivalent.

(1) There exist positive constants c_{1} and c_{2}such that

(A.1)  (E(u, u) + c_{1}||u||_{2}^{2})||u||_{1}^{2/α} ≥ c_{2}||u||_{2}^{2+4/α}

for any u ∈ F ∩ L^{1}(X, d) ∩ F.

(2) T_{t} can be extended to a bounded operator from L^{1}(X, μ) to L^{∞}(X, μ) and there exist c > 0 such that ||T_{t}||_{1→∞} ≤ ct^{−α/2} for any t ∈ (0, 1].

(3) {T_{t}}_{t>0} is ultracontractive and there exists c > 0 such that ||T_{t}||_{2→∞} ≤ ct^{−α/4} for any t ∈ (0, 1].

(A.1) is called the Nash inequality which was introduced in [36]. See [10, 11, 28] for the proof of Theorem A.2.

If μ(X) < +∞, then it is known that the ultracontractivity implies the existence of the heat kernel. The next theorem follows from the results in [11, Section 2.1].

Theorem A.3. Assume that μ(X) < +∞ and that {T_{t}}_{t>0} is ultracontractive. Then there exists p : (0, ∞) × X × X → [0, +∞) such that p ∈ L^{∞}(X^{2}, μ × μ) and

(T_{t}u)(x) = ∫_{X} p(t, x, y)u(y)μ(dy)

for any t > 0, x ∈ X and u ∈ L^{2}(X, μ). p(t, x, y) is called the heat kernel associated with the Dirichlet form (E, F) on L^{2}(X, μ). Moreover, H has compact resolvent, i.e. (H + I)^{−1} is a compact operator. Let (φ_{k})_{k≥1} be a complete orthonormal system of L^{2}(X, μ) consisting of the eigenvalues of H. Assume that Hφ_{k} = λ_{k}φ_{k} and 0 ≤ λ_{k} ≤ λ_{k+1} for any k ≥ 1 and λ_{k} → ∞ as k → ∞. Then φ_{k} ∈ L^{∞}(X, μ) for any k and

(A.2)  p(t, x, y) = ∑_{n=1}^{∞} e^{−λ_{n}t}φ_{n}(x)φ_{n}(y),

where the infinite sum is uniformly convergent on [T, +∞)×X×X for any T > 0. In particular, if φ_{k} ∈ C(X, d), then p(t, x, y) is jointly continuous, i.e. p : (0, 1]×X×X is continuous.

The next theorem gives a sufficient condition for the heat kernel being positive.

Theorem A.4. Assume that μ(X) < +∞, that {T_{t}}_{t>0} is ultracontractive and that (X, d) is arcwise connected. If the heat kernel is jointly continuous and (E, F) is conservative, then p(t, x, y) > 0 for any (t, x, y) ∈ (0, +∞) × X × X.

Proof. Since E(1, 1) = 0, 1 is an eigenfunction of H. Hence by (A.2), p(t, x, x) > 0 for any x ∈ X. Fix x, y ∈ X. Note that if t > s, then

(A.3)  p(t, x, y) = ∫_{X} p(s, x, z)p(t − s, z, y)μ(dy).

Assume that p(s, x, y) > 0. Since p(s, x, y)p(t − s, y, y) > 0, (A.3) implies that p(t, x, y) > 0. Hence there exists t_{∗} ∈ [0, +∞] such that p(t, x, y) = 0 for any t ∈ (0, t_{∗}] and p(t, x, y) > 0 for any t ∈ (t_{∗}, +∞). Next we show that t_{∗} < +∞. Since (X, d) is arcwise connected, there exists γ : [0, 1] → X such that γ is continuous, γ(0) = x and γ(1) = y. For any s ∈ [0, 1], we have an open neighborhood O_{s} of γ(s) that satisfies, p(1, z, w) > 0 for any z, w ∈ O_{s}. Since γ([0, 1]) is compact, there exits {s_{i}}_{i=0}^{m} such that 0 = s_{0} < s_{1} < … < s_{m−1} < s_{m} = 1 and x_{i} ∈ O_{s_{i+1}} for any i = 0, 1, …, m − 1, where x_{i} = γ(s_{i}). By (A.3),

p(m, x, y) = ∫_{X} … ∫_{X} p(1, x, y_{1})p(1, y_{1}, y_{2})…p(1, y_{m−1}, y)μ(dy_{1}) … μ(dy_{m−1}).

Since p(1, x_{i}, x_{i+1}) > 0 for any i = 0, 1, …, m − 1, it follows that p(m, x, y) > 0. Therefore, t_{∗} < m. Now let ℍ_{R} = {z|z ∈ ℂ, Re(z) > 0}. Then the infinite sum

∑_{i=1}^{∞} e^{−λ_{n}z}φ_{n}(x)φ_{n}(y)

is uniformly convergent on ℍ_{R}. Hence p(z, x, y) is extended to a holomorphic function on ℍ_{R}. If t_{∗} > 0, then p(t, x, y) = 0 for any t ∈ (0, t∗]. This implies that p(z, x, y) = 0 for any z ∈ ℍ_{R}. This obviously contradicts the fact that t_{∗} < +∞. Hence t_{∗} = 0. □

Definition A.5. Let X be a set. A pair (E, F) is called a resistance form on X if it satisfies the following conditions (RF1) through (RF5). (RF1) F is a linear subspace of ℓ(X) containing constants and E is a non-negative symmetric quadratic form on F. E(u, u) = 0 if and only if u is constant on X. (RF2) Let ∼ be an equivalent relation on F defined by u ∼ v if and only if u − v is constant on X. Then (F/∼, E) is a Hilbert space. (RF3) For any finite subset V ⊂ X and for any v ∈ ℓ(V), there exists u ∈ F such that u|_{V} = v. (RF4) For any p, q ∈ X,

sup \{\frac{|u(p) − u(q)|^{2}}{E(u, u)} |u ∈ F, E(u, u) > 0\}

is finite. The above supremum is denoted by R(p, q). (RF5) If u ∈ F, then ū ∈ F and E(ū, ū) ≤ E(u, u), where

ū(x) = \begin{cases}1 & if u(x) ≥ 1, \\ u(x) & if 0 ≤ u(x) < 1, \\ 0 & if u(x) < 0. \end{cases}

R(p, q) in the above definition is called the effective resistance between p and q. It is known that R(·, ·) is a distance on X. We call R(·, ·) the resistance metric associated with the resistance form (E, F). See [28] and [29] for more details on resistance forms.

Theorem A.6. Assume that μ(X) < +∞, that {T_{t}}_{t>0} is ultracontractive and that there exist α ∈ (0, 2) and c > 0 such that

(A.4)  ||T_{t}||_{1→∞} ≤ ct^{−α/2}

for any t ∈ (0, 1]. Then, we may choose M > 0 so that

(A.5)  E_{∗}(u, u) ≥ M||u||_{∞}^{2}

for any u ∈ F, where E_{∗}(u, u) = E(u, u) + ||u||_{2}^{2}. In particular, F ⊆ C(X, d) and the heat kernel p(t, x, y) associated with the Dirichlet form (E, F) on L^{2}(X, μ) is jointly continuous. Moreover if (E, F) is conservative and there exists c′ > 0 such that

(A.6)  E(u, u) ≥ c′ ∫_{X}(u − ū)^{2}dμ

for any u ∈ F, where ū = μ(X)^{−1} ∫_{X} udμ, then (E, F) is a resistance form on X. Also if R is the resistance metric associated with (E, F), then (X, R) is bounded.

Proof. Define G_{∗}u = ∫_{0}^{∞} e^{−t}T_{t}udμ. By (A.4), ∫_{0}^{∞} e^{−t}||T_{t}||_{1→∞}dt < +∞. Hence G_{∗} : L^{1}(X, μ) → L^{∞}(X, μ) is a bounded operator. Since G_{∗}φ_{k} = (λ_{k} + 1)^{−1}φ_{k} for any k ≥ 1, we have G_{∗}|_{L^{2}(X,μ)} = (H + I)^{−1}. Note that E_{∗}(G_{∗}u, G_{∗}u) = (u, G_{∗}u) for any u ∈ L^{2}(X, μ). Hence,

||G_{∗}u||_{∗}^{2} ≤ ||u||_{1}||G_{∗}u||_{∞} ≤ M||u||_{1}^{2},

where ||v||_{∗} = \sqrt{E_{∗}(v, v)} and M = ||G_{∗}||_{1→∞}. Now E_{∗}(u, G_{∗}v) = (u, v) for any u ∈ F and any v ∈ L^{2}(X, μ). Therefore

|(v, u)| ≤ |E_{∗}(u, G_{∗}v)| ≤ M||u||_{∗}||G_{∗}u||_{∗} ≤ \sqrt{M}||u||_{∗}||v||_{1}.

Since L^{2}(X, μ) is dense in L^{1}(X, μ), we have u ∈ L^{∞}(X, μ) and (A.5).Since (E, F) is regular, there exist a core C ⊆ F ∩ C_{0}(X, d) such that C is dense in C_{0}(X, d) with respect to || · ||_{∞} and in F with respect to || · ||_{∗}. By (A.5), it follows that F ⊆ C(X, d). Now that φ_{k} ∈ C(X, d), Theorem A.3 shows the continuity of the heat kernel.

Next we will verify the conditions (RF1) - (RF5) to show that (E, F) is a resistance form of X. (RF1) is immediate from the fact that 1 ∈ F, E(1, 1) = 0 and (A.6). By (A.6) and (A.5),

(A.7) (1 + c′)E(u, u) ≥ E(u − ū, u − ū) + ||u − ū||_{2}^{2} = ||u − ū||_{∗}^{2} ≥ M||u − ū||_{∞}^{2}

If F_{∗} = {u|u ∈ F, ū = 0}, then (A.7) says that E and E_{∗} are equivalent on F_{∗}. Since (F, E_{∗}) is complete, (F_{∗}, E) is complete. This implies (RF2). Again by (A.7), there exists c_{1} > 0 such that

c_{1}E(u, u) ≥ c_{1}||u − ū||_{∞}^{2}

for any u ∈ F. Therefore, for any p, q ∈ X and any u ∈ F,

|u(p) − u(q)|^{2} ≤ (|u(p) − ū| + |u(q) − ū|)^{2} ≤ 2c_{1}E(u, u).

Hence

(A.8)  sup \{\frac{|u(p) − u(q)|^{2}}{E(u, u)} |E(u, u) > 0\} ≤ 2c_{1}

for any p, q ∈ X. So we have (RF4). (RF3) holds because F is dense in C_{0}(X, d). (RF5) is immediate from the Markov property of (E, F). Thus we obtain the conditions (RF1) through (RF5). Finally by (A.8), sup_{p,q∈X} R(p, q) ≤ 2c_{1}. □

B. Recurrent case and resistance form

Let (K, S, {F_{i}}_{i∈S}) be a self-similar structure and let d be a metric on K which gives the natural topology of K associated with the self-similar structure. We will consider a resistance from (E, F) on K which satisfies the following conditions (RFA1), (RFA2) and (RFA3): (RFA1) u ∘ F_{i} ∈ F for any i ∈ S. Moreover there exists (r_{i})_{i∈S} ∈ (0, 1)^{S} such that

E(u, v) = ∑_{i∈S} \frac{1}{r_{i}} E(u∘F_{i}, v∘F_{i})

for any u, v ∈ F. (RFA2) Let R be the resistance metric on K associated with (E, F). Then (K, R) is bounded. (RFA3) F ⊆ C(K, d) and F is dense in C(K, d).

Proposition B.1. Under the above situation, R gives the same topology as the one given by d.

Proof. Using the same arguments as in [28, Lemma 3.3.5], we have

(B.1)  r_{w}R(p, q) ≥ R(F_{w}(p), F_{w}(q))

for any w ∈ W_{∗} and any p, q ∈ K.

Let R(x_{n}, x) → 0 as n → ∞. Since (K, d) is compact, there exists x_{∗} ∈ K such that d(x_{n_{i}}, x_{∗}) → 0 as i → ∞ for some {n_{i}}_{i}. Since f ∈ C(K, d) ∩ C(K, R) for any f ∈ F, we see that f(x) = lim_{i∈∞} f(x_{n_{i}}) = f(x_{∗}). Hence x = x_{∗} because (E, F) is a resistance form. This implies that d(x_{n}, x) → 0 as n → ∞.

Conversely, assume lim_{n→∞} d(x_{n}, x) = 0. Define K_{m}(x) = ∪_{w∈W_{m}:x∈K_{w}}K_{w}. Then for any m ≥ 0, x_{n} ∈ K_{m}(x) for sufficiently large n. Hence (B.1) along with (RFA2) implies that R(x_{n}, x) ≤ r_{w}(sup_{p,q∈X} R(p, q)) if w ∈ W_{m} and x ∈ K_{w}. Therefore R(x_{n}, x) → 0 as n → ∞. □

Lemma B.2. Assume (RFA1), (RFA2) and (RFA3). There exists c > 0 such that

(B.2)  E(u, u) ≥ c ∫_{K}(u − (ū)_{μ})^{2}dμ

for any u ∈ F and any elliptic probability measure μ on K, where (ū)_{μ} = ∫_{X} udμ.

Proof. (RFA2) implies that M = sup_{p,q∈K} R(p, q) is finite. Then,

ME(u, u) ≥ R(p, q)E(u, u) ≥ |u(p) − u(q)|^{2}

for any u ∈ F and any p, q ∈ K. Integrating this with respect to p and q, we immediately obtain (B.2). □

Theorem B.3. Let μ be an admissible measure on (K, d). The following two conditions (RE1) and (RE2) are equivalent. (RE1) μ is elliptic. (E, F) is a resistance form on K which satisfies (RFA1), (RFA2) and (RFA3). (RE2) (E, F) is a local regular Dirichlet form on L^{2}(K, μ). 1 ∈ F and E(1, 1) = 0. (E, F, μ) satisfies (SSF) and (PI) and is recurrent.

Moreover if (RE1) or (RE2) holds, then (CHK) and (UPH) are satisfied.

Proof. Note that both (RE1) and (RE2) implies 0 < r_{i} < 1 for any i ∈ S_{∗}. Therefore, if μ is elliptic, than 𝒮_{∗} is elliptic as well.

First we assume (RE1). By [28, Theorem 2.4.2], (E, F) is a regular Dirichlet from on L^{2}(K, μ). To show the local property, suppose that u, v ∈ F and supp(u)∩ supp(v) = ∅. Then we may choose m so that K_{w} ∩ supp(u) ∩ supp(v) = ∅ for any w ∈ W_{m}. Then by (RFA1), E(u, v) = ∑_{w∈W_{m}}(r_{w})^{−1}E(u ∘ F_{w}, v∘F_{w}) = 0. Hence (E, F) has the local property. (SSF) is immediate form (RFA1). (PI) follows from Lemma B.2.

Conversely, assume (RE2). Then by Theorem 3.1.4, we have all the properties required in Theorem A.6. Therefore, we have (RE1).

Finally if (RE2) holds, then by Theorem 3.1.8, we have (CHK) and (UPH). □

C. Heat kernel estimate to the volume doubling property

In this section, (X, d) is a locally compact metric space where every bounded set is precompact, μ is a Radon measure on (X, d) and (E, F) is a local regular Dirichlet form on L^{2}(X, μ). Let H be the non-negative self-adjoint operator on L^{2}(X, μ) associated with (E, F) and let {T_{t}}_{t>0} be the strongly continuous semigroup on L^{2}(X, μ) associated with H.Also let ({X_{t}}_{t>0}, {P_{x}}_{x∈X}) be the diffusion process associated with (E, F). We assume that {T_{t}}_{t>0} is ultracontractive.

Let U be a nonempty open subset of X and let μ_{U} be the restriction of μ on U. Define D_{U} = {u|u ∈ F ∩ C(X), u|_{X\U} ≡ 0}. Let F_{U} be the closure of D_{U} with respect to the inner product E_{∗}(u, v) = E(u, v) + ∫_{X} uvdμ and let E_{U} = E|_{F_{U} ×F_{U}}. By [15, Theorem 4.3], (E_{U}, F_{U}) is a local regular Dirichlet form on L^{2}(U, μ_{U}). Moreover, if ({X_{t}^{U}}_{t>0}, {P_{x}^{U}}_{x∈U}) is the diffusion process associated with (E_{U}, F_{U}) and τ_{U} = inf{t|X_{t} ∉ U}, then

(C.1)  P_{x}^{U}(X_{t}^{U} ∈ A) = P_{x}(X_{t} ∈ A, τ_{U} ≥ t}.

Proposition C.1. Let {T_{t}^{U}}_{t>0} be the strongly continuous semigroup associated with (E, F). Then {T_{t}^{U}}_{t>0} is ultracontractive.

Proof. By (C.1),if u ≥ 0, then

(C.2)  (T_{t}^{U}u)(x) ≤ (T_{t}u)(x)

for μ-a.e. x ∈ X. This immediately shows the desired statement. □

Definition C.2. Let U be a nonempty open subset of X.Define λ_{∗}(U) be

λ_{∗}(U) = inf_{u∈F_{U} ,u≠0} \frac{E_{U}(u, u)}{||u||_{2}^{2}}.

By the variational formula, λ_{∗}(U) is the bottom of the spectrum of −Δ_{U}, where −Δ_{U} is the non-negative self-adjoint operator on L^{2}(U, μ_{U}) associated with (E_{U}, F_{U}).

Theorem C.3. Assume that the heat kernel p(t, x, y) associated with (E, F) is jointly continuous. Suppose that the following two conditions (RFK) and (DUHK) are satisfied for some β > 0: (RFK) There exist r_{∗} > 0 and c_{1} > 0 such that

λ_{∗}(B_{r}(x)) ≤ c_{1}r^{−β}

for any r ∈ (0, r_{∗}] and any x ∈ X. (DUHK) There exist positive constants t_{∗}, c_{2} and c_{3} such that

p(t, x, x) ≤ \frac{c_{2}}{μ(B_{c_{3}t^{1/β}}(x))}

for any t ∈ (0, t_{∗}] and any x ∈ X. Then for any r ∈ (0, min{c_{3}(t_{∗})^{1/β}/3, r_{∗}}],

μ(B_{2r}(x)) ≤ cμ(B_{r}(x)),

where c > 0 is a constant which is independent of x and r.

Proof. Let r ∈ (0, min{c_{3}(t_{∗})^{1/β}/3, r_{∗}}]. For any y ∈ B_{r}(x), (DUHK) implies that

(C.3)  p(c_{∗}r^{β}, y, y) ≤ \frac{c_{2}}{μ(B_{3r}(y))} ≤ \frac{c_{2}}{μ(B_{2r}(x))},

where c_{∗} = (3/c_{3})^{β}. Note that μ(B_{r}(x)) < +∞. Hence by Theorem A.3, there exists a heat kernel p_{B_{r}(x)}(t, y, z) associated with (E_{B_{r}(x)}, F_{B_{r}(x)}).Using (C.2), we see that

p_{B_{r}(x)}(t, y, z) ≤ p(t, y, z)

for μ × μ-a.e. (y, z) ∈ X^{2}. Therefore,

e^{−λ_{∗}(B_{r}(x))t} ≤ ∑_{i≥1} e^{−λ_{i}t} = ∫_{B_{r}(x)^{2}} p_{B_{r}(x)}(t/2, y, z)^{2}μ(dy)μ(dz)

≤ ∫_{B_{r}(x)×X} p(t/2, y, z)^{2}μ(dy)μ(dz) = ∫_{B_{r}(x)} p(t, y, y)μ(dy),

where {λ_{i}}_{i≥1} be the eigenvalues of −Δ_{U}. This and (C.3) along with (RFK) show that

e^{−c_{1}c_{∗}} ≤ e^{−λ_{∗}(B_{r}(x))c_{∗}r^{β}} ≤ c_{2} \frac{μ(B_{r}(x))}{μ(B_{2r}(x))}.

□

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Assumptions, Conditions and Properties in Parentheses

(A)_{n}, 17 (AS1), 28 (AS2), 28 (AS3), 28 (CHK), 61 (D1), 45 (D2), 45 (DLHK), 64 (DUHK), 63, 88 (EL1), 12 (EL2), 12 (ELm), 14 (ELmg), 19 (G1), 11 (G2), 11 (GE), 17 (GF1), 76 (GF2), 76 (GF3), 76 (GF4), 76 (GSC1), 70 (GSC2), 70 (GSC3), 70 (GSC4), 70 (LF), 17 (LHK), 64 (M1), 14 (M2), 14 (M3), 14 (P1), 3 (P2), 3 (P3), 3 (PI), 60 (R1), 31 (R2), 31 (RE1), 87 (RE2), 87 (RF1), 84 (RF2), 84 (RF3), 84 (RF4), 85 (RF5), 85 (RFA1), 86 (RFA2), 86 (RFA3), 86 (RFK), 88 (S1), 10 (S2), 10 (SC1), 36 (SC2), 36 (SSF), 59 (SSF1), 59 (SSF2), 59 (UHK), 64 (UPH), 61 (VD), 17 (V D)_{n}, 17 (VDd), 57

List of Notations

A_{X,x}(w), 28 C_{w}, 10 D_{𝒮}, 44 GSC(n, l, s), 70 h_{A}, 61 K(Γ), 16 K[X], 23 K^{(n)}(Γ, A), 16 K_{s}(x), 16 K_{w}[X], 23 L_{w}, 10 M_{A,τ,a}, 66 n_{A}(𝒮), 55 N_{X,x}(w), 29 O_{Σ_{0},x}(ω), 25 p_{U}(t, x, y), 62, 74 Q_{m}, 24 R(w, v, R), 48 rf_{k}, 70 R_{w}, 10 S_{k,s}, 70 U_{s}(x), 16 U_{s}^{(n)}(x), 16 V_{0}, 13 W (Γ, A), 16 W^{(n)}(Γ, A), 16 W^{1,2}(K), 7 W_{∗}(S), 9 W_{m}(S), 9 W_{#}(S), 9 #(·), 13 E_{U}, 62, 74 F_{U}, 62, 74 ℓ(V), 9 A, 47 CH, 43 CH_{m}(x, y), 66 CH(x, y), 43 C_{L}, 13 E𝒮(Σ), 22 IP(L), 47 IP(L, 𝒮), 47 IP(L, 𝒮, R), 48 IT (L), 47 IT (L, 𝒮), 47 IT (L, 𝒮, R), 48 M(K), 14 M_{1}(K), 14 M_{VD}(L, 𝒮), 22 P_{L}, 13 R_{1}, 33 R_{2}, 33 R_{L}, 25 𝒮(a), 13 𝔖_{LF}(Σ, L), 33 𝔖(Σ), 13 δ^{(n)}(·, ·), 53 Δ_{U}, 62, 74 ι_{X}^{w}, 24 Λ_{s}(a), 13 Λ_{s,w}, 16 Λ_{s,w}^{R}, 30 Λ_{s,x}, 16 Λ_{s,x}^{n}, 16 Φ_{k,s}, 70 Ψ_{k,l}, 70 ρ_{m}, 23 ρ_{m,n}, 24 Σ(S), 9 Σ_{w}[X], 23 Σ[X], 23 σ_{i}, 9

∼_{GE}, 22

(·, ·)_{V}, 9

∼_{n}, 50

Index

adapted, 52

n-, 53 arithmetic, 63

chain, 43 conservative, 60 corresponding pair, 25 critical set, 13

diamond fractal, 39

effective resistance, 85 elliptic

measure, 14 scale, 12

empty word, 9

gauge function, 11 induced by measure, 16 of scale, 11 self-similar, 13

generalized Sierpinski carpet, 70 generator

of relations, 26 gentle, 17

among scales, 21 geodesic, 63

distance, 63 pair, 63

Green function, 76

hitting time, 61

independent, 23 intersection pair, 47 intersection type, 47

finite, 47 irreducible, 66

k-neighbors, 70

length of a word, 9 L-isomorphism, 49 locally finite, 17 L-similar, 50 L-similitude, 50

modified Sierpinski gasket, 68 n-adapted, 53 Nash inequality, 83 near diagonal lower estimate, 81

partition, 9

Poincaré inequality, 60 post critical set, 13 post critically finite, 34 pseudodistance, 43

associated with a scale, 44

qdistance, 51 quasidistance, 51

rationally ramified, 26

recurrent, 59

harmonic structure, 65 recursive system, 66 refinement, 10 relation, 25

generated by, 26 relation matrix, 66 relation set, 26 resistance form, 84 resistance metric, 85 resistance scaling ratio, 59 right continuous scale, 11

scale, 10 elliptic, 12 induced by gauge function, 11 right continuous, 11 self-similar, 13

self-similar Dirichlet form, 59 gauge function, 13 measure, 15 scale, 13 set, 13

self-similar structure, 13 strongly finite, 13

shift map, 9 space, 9 Sierpinski carpet

generalized, 70 Sierpinski cross, 37 Sierpinski gasket, 26 sub-relation, 26

ultracontractive, 83 uniform positivity of hitting time, 61

volume doubling property with respect to scale, 17

weakly symmetric, 6 word

empty, 9 length of, 9

word space, 9