On the Lipschitz equivalence of self-affine sets

TL;DR

This paper investigates the Lipschitz equivalence of self-affine sets generated by expanding integer matrices and digit sets, establishing conditions under which these sets are Lipschitz equivalent based on their $w$-Hausdorff dimension.

Contribution

It introduces a pseudo-norm based hyperbolic graph construction and extends known self-similar set results to self-affine sets under the open set condition.

Findings

Self-affine sets can be identified with hyperbolic boundaries.

Lipschitz equivalence is characterized by equal $w$-Hausdorff dimensions.

Extends results from self-similar to self-affine sets.

Abstract

Let $A$ be an expanding $d\times d$ matrix with integer entries and ${\mathcal D}\subset {\mathbb Z}^d$ be a finite digit set. Then the pair $(A, {\mathcal D})$ defines a unique integral self-affine set $K=A^{-1}(K+{\mathcal D})$. In this paper, by replacing the Euclidean norm with a pseudo-norm $w$ in terms of $A$, we construct a hyperbolic graph on $(A, {\mathcal D})$ and show that $K$ can be identified with the hyperbolic boundary. Moreover, if $(A, {\mathcal D})$ safisfies the open set condition, we also prove that two totally disconnected integral self-affine sets are Lipschitz equivalent if an only if they have the same $w$-Hausdorff dimension, that is, their digit sets have equal cardinality. We extends some well-known results in the self-similar sets to the self-affine sets.

Equations

$$K=\bigcup _{i=1}^N{S}_i(K).$$

$$K=A^{-1}(K+\mathcal{D})$$

$$C^{-1}{d}_1(x,y)\le d_2(\sigma (x),\sigma (y))\le C{d}_1(x,y),\forall x,y\in E.$$

$$C^{-1}\Vert x-y\Vert \le \Vert \sigma (x)-\sigma (y)\Vert \le C\Vert x-y\Vert ,\forall x,y\in E,$$

$$C^{-1}\Vert x-y{\Vert }^{1/\eta }\le \Vert \sigma (x)-\sigma (y)\Vert \le C\Vert x-y{\Vert }^{\eta },\forall x,y\in E.$$

$${\mathcal{E}}_h=\bigcup _{n=1}^{\infty }\{(\mathbf{u},\mathbf{v})\in X_n\times X_n:\mathbf{u}\ne \mathbf{v},\text{ and }{S}_{\mathbf{u}}(J)\cap S_{\mathbf{v}}(J)\ne \varnothing \}.$$

$$C^{-1}w(\varphi (\xi )-\varphi (\eta ))\le {\rho }_a(\xi ,\eta {)}^{\alpha }\le Cw(\varphi (\xi )-\varphi (\eta )),\forall \xi \ne \eta \in \partial X$$

$$|x\wedge y|=\frac{1}{2}\big{(}|x|+|y|-d(x,y)\big{)}$$

$$|x\wedge y|\ge \min \{|x\wedge z|,|z\wedge y|\}-\delta \forall x,y,z\in X.$$

$${\rho }_a(x,y)={\delta }_{x,y}\exp (-a|x\wedge y|),$$

$$(x,y)\in {\mathcal{E}}_h\Rightarrow |x|=|y|,\text{mbox}andeither{x}^{-}=y^{-}\text{mbox}or(x^{-},y^{-})\in {\mathcal{E}}_h.$$

$$\pi (x,y)=\pi (x,u)\cup \pi (u,v)\cup \pi (v,y)$$

$$|x\wedge y|=h-\ell /2$$

$$w(x)=\sum _{n=-\infty }^{\infty }{q}^{-n/d}h(A^{n}x),x\in {\mathbb{R}}^d.$$

$$C_{ϵ}^{-1}\Vert x{\Vert }^{\ln q/d\ln ({\lambda }_0-ϵ)}\le w(x)\le C_{ϵ}\Vert x{\Vert }^{\ln q/d\ln ({\lambda }_1+ϵ)}.$$

$$\frac{\ln q}{d\ln {\lambda }_1}{\dim }_H^{w}E\le {\dim }_{H}E\le \frac{\ln q}{d\ln {\lambda }_0}{\dim }_H^{w}E.$$

$${\mathcal{E}}_h=\bigcup _{n=1}^{\infty }\{(\mathbf{u},\mathbf{v})\in X_n\times X_n:\mathbf{u}\ne \mathbf{v},\text{ and }{J}_{\mathbf{u}}\cap J_{\mathbf{v}}\ne \varnothing \}.$$

$$g(x)=A^{n-m}x+\mathbf{d},\text{where }\mathbf{d}\in {\mathbb{Z}}^d$$

$$\{[T_i]:1\le i\le n\}=\{[T_i^{\prime }]:1\le i\le n^{\prime }\}$$

$$g\circ S_{{\mathbf{u}}_{i}j}=g\circ S_{{\mathbf{u}}_i}\circ S_j=S_{{\mathbf{u}}_i^{\prime }}\circ S_j=S_{{\mathbf{u}}_i^{\prime }j}.$$

$$\{g\circ S_{{\mathbf{u}}_{i}j}:{\mathbf{u}}_i\in T,j\in \Sigma \}=\{S_{{\mathbf{u}}_i^{\prime }j}:{\mathbf{u}}_i^{\prime }\in T^{\prime },j\in \Sigma \}.$$

$$J_{\mathbf{u}}\cap J_{\mathbf{v}}=\varnothing ⟹{dist}_w(J_{\mathbf{ui}},J_{\mathbf{vj}})\ge c{q}^{-n/d},\forall \mathbf{i},\mathbf{j}\in {\Sigma }^k.$$

$$K_{\mathbf{u}}\cap K_{\mathbf{v}}=\varnothing ⟹{dist}_w(K_{\mathbf{u}},K_{\mathbf{v}})\ge c^{\prime }{q}^{-n/d}.$$

$$c^{\prime }=\inf \{w(x):x\in \cup \{K-K+d:d\in {\mathbb{Z}}^d\text{ such that }0\notin K-K+d\}\}.$$

$$w(S_{\mathbf{u}}(x)-S_{\mathbf{v}}(y))=w(A^{-n}(x-y+d_{\mathbf{u}}-d_{\mathbf{v}}))=q^{-n/d}w(x-y+d_{\mathbf{u}}-d_{\mathbf{v}})>0.$$

$$w(S_{\mathbf{u}}(x)-S_{\mathbf{v}}(y))\ge c^{\prime }{q}^{-n/d}.$$

$${dist}_w(J_{\mathbf{ui}},J_{\mathbf{vj}})$$

$$\ge c^{\prime }{q}^{-n/d}-(2c^{\prime }/3)q^{-n/d}\ge (c^{\prime }/3)q^{-n/d}\forall \mathbf{i},\mathbf{j}\in {\Sigma }^k.$$

$$\#\{\mathbf{u}\in X:(J+d_{\mathbf{u}})\cap V\ne \varnothing \}\le \gamma ,$$

$$C^{-1}w(\varphi (\xi )-\varphi (\eta ))\le {\rho }_a(\xi ,\eta {)}^{\alpha }\le Cw(\varphi (\xi )-\varphi (\eta )),\forall \xi \ne \eta \in \partial X$$

$$\#\{\mathbf{v}\in X:|\mathbf{v}|=n-1,n\text{ or }n+1,J_{\mathbf{v}}\cap J_{\mathbf{u}}\ne \varnothing \}\le \gamma (b{q}^{\frac{1}{d}})+\gamma (b)+\gamma (b{q}^{-\frac{1}{d}}).$$

Full text

The research is supported in part by the NNSF of China (No.11301322), the Fundamental and Frontier Research Project of Chongqing (No.cstc2015jcyjA00035)

1. Introduction

Let $M_{d}({\mathbb{Z}})$ denote the class of $d\times d$ matrices with integer entries and let $A\in M_{d}({\mathbb{Z}})$ be expanding, i.e., all its eigenvalues in moduli are strictly bigger than $1$. Let ${\mathcal{D}}=\{d_{1},\dots,d_{N}\}\subset{\mathbb{Z}}^{d}$ be a digit set. Define affine maps $S_{i}(x)=A^{-1}(x+d_{i})$ for all $i$. Then $\{S_{i}\}_{i=1}^{N}$ forms an iterated function system (IFS). There exists a unique nonempty compact set $K$ in ${\mathbb{R}}^{d}$ [6] satisfying

[TABLE]

$K$ is called an (integral) self-affine set, and a self-similar set if $A$ is a similar matrix (i.e., $A=nI$ where $n\in{\mathbb{N}}$ and $I$ is an identity matrix). Usually, we also write it as

[TABLE]

when we emphasize the affine pair $(A,{\mathcal{D}})$. The IFS or the $(A,{\mathcal{D}})$ is said to satisfy the open set condition (OSC) if there exists a bounded nonempty open set $O$ such that $\cup_{i=1}^{N}S_{i}(O)\subset O$ and $S_{i}(O)\cap S_{j}(O)=\emptyset$ for $i\neq j$. The McMullen-Bedford sets ([2],[19]) are special cases of such self-affine sets (see Figure 1).

There are many studies on self-affine sets (see book [6]). Moreover, the related self-affine tiles and tilings are also hot topics in the literature (see the survey paper [25] and references therein).

Two metric spaces $(E,d_{1})$ and $(F,d_{2})$ are said to be Lipschitz equivalent, denote by $E\simeq F$, if there exists a bi-Lipschitz map $\sigma:E\to F$, i.e., $\sigma$ is a bijection and there exists a constant $C>0$ such that

[TABLE]

If $E,F\subset{\mathbb{R}}^{d}$ and $E\simeq F$, then the above inequality becomes

[TABLE]

where $\|\cdot\|$ is the Euclidean norm. The Hausdorff dimension is an invariant under the bi-Lipschitz map. Like the topological equivalence, Lipschitz equivalence is an important tool for the classification of fractals in fractal geometry and geometric measure theory ([7],[4],[27],[15]). The study of Lipschitz equivalence on Cantor sets was initiated by Copper and Pignataro [3] and Falconer and Marsh [8]. Along this line, it has been undergoing a great development by many people ([5],[16],[18],[20]-[22],[28]-[30]). Up to now, the following is an elegant result on the Lipschitz equivalence of self-similar sets.

Theorem 1.1** ([20],[28],[18],[29]).**

Let $A=nI$ be a similar matrix and $K=A^{-1}(K+{\mathcal{D}}_{1}),K^{\prime}=A^{-1}(K^{\prime}+{\mathcal{D}}_{2})$ be two self-similar sets as in (1.1). If both the IFSs satisfy the OSC and $K,K^{\prime}$ are totally disconnected. Then $K\simeq K^{\prime}$ if and only if $\#{\mathcal{D}}_{1}=\#{\mathcal{D}}_{2}$.

However, due to the complexity and non-uniform contractility from the matrix A, it is difficult to investigate the geometric and topological properties of self-affine sets. To our knowledge, there are very few results on the Lipschitz equivalence of self-affine sets. For example, even if $K,K^{\prime}$ in Figure 1 have the same Hausdorff dimension, it is still not clear whether they are Lipschitz equivalent or not.

In order to absorb the non-uniform contractility from $A$, He and Lau [10] introduced a concept of pseudo-norm $w$ in terms of $A$ (see Section 2) to replace the Euclidean norm and defined the (generalized) $w$-Hausdorff measure ${\mathcal{H}}_{w}^{s}$, and $w$-Hausdorff dimension $\dim_{H}^{w}$. Moreover, they extended Schief’s well-known result on self-similar sets [23] to self-affine sets.

In this paper, we mainly apply the pseudo-norm approach to make an attempt on the Lipschitz equivalence of self-affine sets. For distinction, we call $E,F$ $w$-Lipschitz equivalent, and denote by $E\simeq_{w}F$ if we replace $\|\cdot\|$ by $w(\cdot)$ in (1.2).

On the other hand, Falconer and Marsh in [8] proposed a nearly Lipschitz equivalence between $E$ and $F$, denote by $E\simeq_{n}F$, in the sense that for any $0<\eta<1$ there exist a bijective map $\sigma:E\to F$ and $C>0$ such that

[TABLE]

The Hausdorff dimension is also an invariant under nearly Lipschitz equivalence. A relationship between the two kinds of Lipschitz equivalence is as follows.

Proposition 1.2**.**

Suppose the eigenvalues of $A$ have equal moduli. If $E\simeq_{w}F$ then $E\simeq_{n}F$.

In studying the Lipschitz equivalence of self-similar sets, the author developed a technique of augmented tree (refer to a series of papers [18],[5],[16]). An augmented tree is defined on the symbolic space of the self-similar IFS by adding more edges, and it is a Gromov hyperbolic graph ([13],[14]).

Now under the setting of self-affine IFS as in (1.1), we let $\Sigma=\{1,\dots,N\}$ and $\Sigma^{*}:=\bigcup_{n=0}^{\infty}\Sigma^{n}$ where $\Sigma^{0}=\emptyset$. For ${\bf u}=i_{1}\cdots i_{n}\in\Sigma^{*}$, write the length $|{\bf u}|=n$ and the composition $S_{\bf u}=S_{i_{1}}\circ\cdots\circ S_{i_{n}}$. Say ${\bf u},{\bf v}\in X$ are equivalent, denote by ${\bf u}\sim{\bf v}$, if $S_{{\bf u}}=S_{{\bf v}}$. Then $\sim$ determines an equivalence relation on $\Sigma^{*}$. Set $X_{n}=\Sigma^{n}/\sim$ for each $n$. Then $X=\bigcup_{n=0}^{\infty}X_{n}$ is the quotient space of $\Sigma^{*}$ and $[{\bf u}]$ the equivalence class. For convenience, we still use ${\bf u}\in X$ to replace $[{\bf u}]\in X$ with no confusions.

There is a natural graph structure on $X$ by the standard concatenation of words (see details in Section 2), we denote the edge set by ${\mathcal{E}}_{v}$. Let $J$ be a nonempty bounded closed invariant set of the IFS, i.e., $S_{i}(J)\subset J$ for each $i$. We define horizontal edges on the graph $(X,{\mathcal{E}}_{v})$ by

[TABLE]

Let ${\mathcal{E}}={\mathcal{E}}_{v}\cup{\mathcal{E}}_{h}$, then the graph $(X,{\mathcal{E}})$ resembles the augmented tree (see Definition 2.1).

We use the standard notation on hyperbolic graph $X$ introduced by Gromov ([9], [26]). The hyperbolic boundary is defined by $\partial X=\hat{X}\setminus X$ where $\hat{X}$ is the completion of $X$ under a visual metric $\rho_{a}$ on $X$ (see Section 2). The following main result is a generalization of the self-similar case ([13],[24],[14]).

Theorem 1.3**.**

Let $K$ be the integral self-affine set as in (1.1) with $|\det A|=q$. Then the graph $(X,{\mathcal{E}})$ is hyperbolic. Moreover, $K$ is Hölder equivalent to the hyperbolic boundary of $(X,{\mathcal{E}})$, i.e., there exists a bijective map $\phi:\partial X\to K$ such that

[TABLE]

where $\alpha=\frac{\log q}{da}$ and $C>0$ is a constant.

The theorem together with Theorem 1.1 of [5] (see Theorem 3.4) helps us extend Theorem 1.1 to the framework of self-affine sets.

Theorem 1.4**.**

Let $K=A^{-1}(K+{\mathcal{D}}_{1})$ and $K^{\prime}=A^{-1}(K^{\prime}+{\mathcal{D}}_{2})$ be two integral self-affine sets. Suppose both the IFSs satisfy the OSC and $K,K^{\prime}$ are totally disconnected. Then $K\simeq_{w}K^{\prime}$ if and only if $\#{\mathcal{D}}_{1}=\#{\mathcal{D}}_{2}$.

As a corollary, if we further assume that the eigenvalues of $A$ have equal moduli then $K,K^{\prime}$ are nearly Lipschitz equivalent if and only if $\#{\mathcal{D}}_{1}=\#{\mathcal{D}}_{2}$ (Corollary 4.7). Moreover, we shall show in Example 4.9 that the McMullen-Bedford sets in Figure 1 are $w$-Lipschitz equivalent.

In general, the IFS in (1.1) has overlaps. It actually satisfies the weak separation condition ([12]) which is weaker than the OSC. Hence if we remove the OSC in the assumption, then the theorem would be false. As for the overlapping case, we need more deep discussions (see [16]).

For the organization of the paper, we recall basic results on hyperbolic graph and pseudo-norm in Section 2. In Section 3, we construct a hyperbolic graph on the affine pair $(A,{\mathcal{D}})$ and prove Theorem 1.3. In Section 4, we first prove Proposition 1.2, and then prove Theorem 1.4 by some technical lemmas.

2. Hyperbolic graph and pseudo-norm

Let $X$ be an infinite connected graph. For $x,y\in X$, let $\pi(x,y)$ denote a geodesic from $x$ to $y$, and $d(x,y)$ its length. Fix a vertex $o$ as a reference point of $X$, and let $|x|=d(o,x)$. The degree of a vertex $x$ is the total number of edges connecting to $x$ and is denoted by $\deg(x)$. A graph is said to be of bounded degree if $\max\{\deg(x):x\in X\}<\infty$. According to [26], for $x,y\in X$, let

[TABLE]

denote the Gromov product, and call $X$ hyperbolic if there is $\delta\geq 0$ such that

[TABLE]

For $a>0$ with $\exp(3\delta a)<\sqrt{2}$, we define a binary map on $X$ by

[TABLE]

where $\delta_{x,y}=0,1$ according to $x=y$ or $x\neq y$. The $\rho_{a}$ is not necessarily a metric, but it is equivalent to a metric ([26]). Hence we always regard $\rho_{a}$ as a visual metric for convenience. Let $\hat{X}$ be the $\rho_{a}$-completion of $X$. We call $\partial X=\hat{X}\setminus X$ the hyperbolic boundary of $X$. It is clear that $\rho_{a}$ can be extended to $\partial X$, and $\partial X$ is a compact set under $\rho_{a}$. It is useful to identify $\xi\in\partial X$ with a geodesic ray in $X$ that converges to $\xi$.

Let $X$ be a tree (i.e., any two distinct vertices can be connected by only one path). Trivially, $X$ is hyperbolic (with $\delta=0$), and the hyperbolic boundary is a Cantor set. We use ${\mathcal{E}}_{v}$ to denote the set of edges of $X$ ($v$ for vertical), and $X_{n}=\{x\in X:|x|=n\}$ the $n$-th level of $X$. We introduce some additional edges on each level of $X$.

Definition 2.1** ([11],[13]).**

Let $(X,{\mathcal{E}}_{v})$ be a tree. We call $(X,{\mathcal{E}})$ an augmented tree if ${\mathcal{E}}={\mathcal{E}}_{v}\cup{\mathcal{E}}_{h}$, where ${\mathcal{E}}_{h}\subset(X\times X)\setminus\{(x,x):\ x\in X\}$ is symmetric and satisfies

[TABLE]

($x^{-}$ is the predecessor of $x$.) We call elements in ${\mathcal{E}}_{h}$ horizontal edges.

Furthermore, if each vertex of $X$ has $N$ offspring, we call $(X,{\mathcal{E}}_{v})$ an $N$-ary tree and $(X,{\mathcal{E}})$ an $N$-ary augmented tree.

For $x,y\in X$, the geodesic path of $x,y$ is not unique in general, but there is a canonical one of the form

[TABLE]

where $\pi(x,u),\pi(v,y)$ are vertical paths, $\pi(u,v)$ is a horizontal path, and for any geodesic $\pi^{\prime}(x,y)$, $d(o,\pi(u,v))\leq d(o,\pi^{\prime}(x,y))$. It can happen that there are only two parts with $v=y$ or $x=u$. For a canonical geodesic $\pi(x,y)$, the Gromov product can be written as

[TABLE]

where $h,\ell$ are the level and the length of the horizontal part $\pi(u,v)$, respectively.

Let $A\in M_{d}({\mathbb{Z}})$ be expanding with $|\det A|=q$. Let $B_{\delta}:=B(0,\delta)$ be the open ball in ${\mathbb{R}}^{d}$ with center at [math] and radius $\delta$. Following [10], we introduce the notion of pseudo-norm (with respect to $A$) as follows: For $0<\delta<1/2$, let $\varphi\geq 0$ be a $C^{\infty}$ function supported in $B_{\delta}$ with $\varphi(x)=\varphi(-x)$ and $\int_{{\mathbb{R}}^{d}}\varphi=1$. Let $V=AB_{1}\setminus B_{1}$, and let $h=\chi_{V}\ast\varphi$ be the convolution of the indicator function $\chi_{V}$ and $\varphi$. We define

[TABLE]

Then $w(x)$ satisfies

1. (1)

$w(x)=w(-x)$ and $w(x)=0$ if and only if $x=0$, 2. (2)

$w(Ax)=q^{1/d}w(x)$, and 3. (3)

there exists $\beta>0$ such that $w(x+y)\leq\beta\max\{w(x),w(y)\}$.

The $w$ is used as a distance (ultra-metric) to replace the Euclidean distance to define $\hbox{diam}_{w}(E)=\sup\{w(x-y):x,y\in E\}$ (the diameter of a set $E$) and $\hbox{dist}_{w}(E,F)=\inf\{w(x-y):x\in E,y\in F\}$ (the distance between sets $E$ and $F$). Moreover, the $w$-Hausdorff distance $d^{w}_{H}$, $w$-Hausdorff measure ${\mathcal{H}}_{w}^{s}$, and $w$-Hausdorff dimension $\dim_{H}^{w}$ are also well-defined accordingly.

The new and old definitions of norm and dimension have a simple relationship through $\lambda_{1}$ and $\lambda_{0}$, the maximal and minimal moduli of the eigenvalues of $A$.

Proposition 2.2** ([10]).**

(i) If $\|x\|\leq 1$. Then for any $0<\epsilon<\lambda_{0}-1$, there exists a constant $C_{\epsilon}>0$ such that

[TABLE]

(ii) For any subset $E$ of ${\mathbb{R}}^{d}$, we have

[TABLE]

Under the pseudo-norm, most of the basic properties for the self-similar sets (including Schief’s basic result on the OSC) can be carried to the self-affine sets and graph-directed sets (see [10] and [17]).

Theorem 2.3** ([10]).**

Let $A$ be an expanding matrix with $|\det A|=q$ and ${\mathcal{D}}=\{d_{1},\dots,d_{N}\}\subset{\mathbb{R}}^{d}$. Let $K=A^{-1}(K+{\mathcal{D}})$ be the associated self-affine set. If $(A,{\mathcal{D}})$ satisfies the OSC, then $\dim^{w}_{H}K=d\ln N/\ln q:=s$ and $0<{\mathcal{H}}_{w}^{s}(K)<\infty$.

3. Hyperbolic graph induced by $(A,{\mathcal{D}})$

Let the IFS $\{S_{i}\}_{i=1}^{N}$ and the self-affine set $K$ be as in (1.1). Now we construct a graph structure on the symbolic space that represents the IFS. Let $\Sigma=\{1,\dots,N\}$ and $\Sigma^{*}:=\bigcup_{n=0}^{\infty}\Sigma^{n}$ be the symbolic space where $\Sigma^{0}=\emptyset$ (as a reference point). For ${\bf u}=i_{1}\cdots i_{n}\in\Sigma^{*}$, write the length $|{\bf u}|=n$ and the composition $S_{\bf u}=S_{i_{1}}\circ\cdots\circ S_{i_{n}}$. Say ${\bf u},{\bf v}\in X$ are equivalent, denote by ${\bf u}\sim{\bf v}$, if $S_{{\bf u}}=S_{{\bf v}}$. Then $\sim$ defines an equivalence relation on $\Sigma^{*}$. Let $X_{n}=\Sigma^{n}/\sim$ for each $n$. Then $X=\bigcup_{n=0}^{\infty}X_{n}$ is the quotient space of $\Sigma^{*}$ and $[{\bf u}]$ is the equivalence class containing ${\bf u}$. For convenience, we still use ${\bf u}\in X$ to replace $[{\bf u}]\in X$ with no confusions.

There is a natural graph structure on $X$ by the standard concatenation of finite words, we denote the edge set by ${\mathcal{E}}_{v}$. That is, $({\bf u},{\bf v})\in{\mathcal{E}}_{v}$ if and only if there exist ${\bf i}\in[{\bf u}],{\bf j}\in[{\bf v}]$ and some $\ell\in\Sigma$ such that ${\bf j}={\bf i}\ell$ or ${\bf i}={\bf j}\ell$. We notice that if the IFS satisfies the OSC, then $S_{{\bf u}}\neq S_{\bf v}$ for any distinct ${\bf u},{\bf v}$. Hence $X=\Sigma^{*}$ and $(X,{\mathcal{E}}_{v})$ is an $N$-ary tree.

Let $J$ be a closed invariant set of the IFS, i.e., $S_{i}(J)\subset J$ for all $i$. For ${\bf u}=i_{1}\cdots i_{n}\in\Sigma^{*}$, we let $J_{{\bf u}}:=S_{{\bf u}}(J)=A^{-n}(J+d_{{\bf u}})$ where $d_{{\bf u}}=d_{i_{n}}+Ad_{i_{n-1}}+\cdots+A^{n-1}d_{i_{1}}$. According to the geometry of $K$, we define more (horizontal) edges on $X$:

[TABLE]

Let ${\mathcal{E}}={\mathcal{E}}_{v}\cup{\mathcal{E}}_{h}$, then the graph $(X,{\mathcal{E}})$ resembles the augmented tree in Definition 2.1 by the observation: As $J_{\bf u}\subset J_{{\bf u}^{-}},J_{\bf v}\subset J_{{\bf v}^{-}}$, if $({\bf u},{\bf v})\in{\mathcal{E}}_{h}$ then either ${\bf u}^{-}={\bf v}^{-}$ or $({\bf u}^{-},{\bf v}^{-})\in{\mathcal{E}}_{h}$.

Lemma 3.1** ([24]).**

The graph $(X,{\mathcal{E}})$ is hyperbolic if and only if the lengths of horizontal geodesics are uniformly bounded.

In particular, if the IFS satisfies the OSC, then the graph $(X,{\mathcal{E}})$ indeed is an $N$-ary augmented tree which has been studied in detail in [13],[18],[5] and [14].

The invariant set $J$ can be quite flexible, for example we can take $J=K$, or take $J=\overline{U}$ for the open set $U$ in the OSC, or take $J$ to be some sufficiently large closed ball. The graph $(X,{\mathcal{E}})$ depends on the choice of $J$. But under our IFS as in (1.1), the hyperbolic boundary is the same as they can be identified with the underlying self-affine set (see Theorem 3.7).

Now we fix $J=\overline{B_{\delta}}$ (the closure of a ball $B_{\delta}$). For $T\subset X$, we say that $T$ is a horizontal component if $T\subset X_{n}$ for some $n$ and $T$ is a maximal connected subset of $X_{n}$ with respect to ${{\mathcal{E}}}_{h}$. Write $J_{T}:=\bigcup_{{\bf u}\in T}S_{{\bf u}}(J)$. Geometrically, $T$ is a horizontal component of $X$ if and only if $J_{T}$ is a connected component of $\bigcup_{{\bf u}\in X_{n}}S_{{\bf u}}(J)$.

Let $T\subset X_{n},T^{\prime}\subset X_{m}$ be two horizontal components of $X$. We say that $T$ and $T^{\prime}$ are equivalent, denote by $T\sim T^{\prime}$, if there exists an affine map

[TABLE]

such that $\{g\circ S_{\bf u}:{\bf u}\in T\}=\{S_{{\bf u}^{\prime}}:{\bf u}^{\prime}\in T^{\prime}\}$. Obviously, if $T\sim T^{\prime}$, then $\#T=\#T^{\prime}$ and $g(J_{T})=J_{T^{\prime}}$. Denote by $[T]$ the equivalence class and ${\mathcal{F}}$ the family of all horizontal components of $X$.

Proposition 3.2**.**

Let $T,T^{\prime}\in{\mathcal{F}}$, and let $\{T_{1},\dots,T_{n}\},\{T_{1}^{\prime},\dots,T_{n^{\prime}}^{\prime}\}\subset{\mathcal{F}}$ be the horizontal components that consist of offspring of $T,T^{\prime}$ respectively. Suppose $T\sim T^{\prime}$. Then

[TABLE]

counting multiplicity. In particular, $n=n^{\prime}$.

Proof.

Since $T\sim T^{\prime}$, without loss of generality, we assume that $T=\{{\bf u}_{1},\dots,{\bf u}_{m}\},T^{\prime}=\{{\bf u}^{\prime}_{1},\dots,{\bf u}^{\prime}_{m}\}$ and $g\circ S_{{\bf u}_{i}}=S_{{\bf u}^{\prime}_{i}}$ where $1\leq i\leq m$ and $g$ is an affine map as in the definition. Then for any $1\leq i\leq m$ and $j\in\Sigma$,

[TABLE]

Hence

[TABLE]

It follows that $S_{{\bf u}_{i}k}=S_{{\bf u}_{j}\ell}$ if and only if $S_{{\bf u}_{i}^{\prime}k}=S_{{\bf u}_{j}^{\prime}\ell}$ and $S_{{\bf u}_{i}k}(J)\cap S_{{\bf u}_{j}\ell}(J)\neq\emptyset$ if and only if $S_{{\bf u}_{i}^{\prime}k}(J)\cap S_{{\bf u}_{j}^{\prime}\ell}(J)\neq\emptyset$, completing the proof. ∎

Definition 3.3**.**

We call the graph $(X,{\mathcal{E}})$ simple if the equivalence classes in ${\mathcal{F}}$ is finite, that is, $\#({\mathcal{F}}/\sim)<\infty$.

We remark that the definition of simple graph is slightly stronger than the original one in [5] which is defined from the graphical point of view. Hence under the OSC (where the graph $(X,{\mathcal{E}})$ becomes an $N$-ary augmented tree), we have

Theorem 3.4** ([5]).**

Suppose an $N$-ary augmented tree $(X,{\mathcal{E}})$ is simple, then

(i) $(X,{\mathcal{E}})$ is a hyperbolic graph;

(ii) $\partial(X,{\mathcal{E}})\simeq\partial(X,{\mathcal{E}}_{v})$, which is an $N$-Cantor set.

Lemma 3.5**.**

Let $K$ be an integral self-affine set as in (1.1). Then for any bounded closed invariant set $J$, there exist $c>0$ and $k\geq 0$ such that for any $n\geq 1$ and ${\bf u,v}\in\Sigma^{n}$,

[TABLE]

Proof.

We first claim that there exists $c^{\prime}>0$ such that for any integer $n\geq 1$ and ${\bf u},{\bf v}\in\Sigma^{n}$,

[TABLE]

Indeed let

[TABLE]

Then $c^{\prime}>0$ as $K$ is compact. If $K_{\bf u}\cap K_{\bf v}=\emptyset$, then for any $x,y\in K$, we have

[TABLE]

By making use of $d_{\bf u},d_{\bf v}\in{\mathbb{Z}}^{d}$ and the above expression of $c^{\prime}$, hence we have

[TABLE]

For the invariant set $J$, we have $K\subset J$ and the $w$-Hausdorff distance $d_{H}^{w}(K_{\bf i},J_{\bf i})\leq c_{1}q^{{-k}/d}$ for all ${\bf i}\in\Sigma^{k}$. In particular we take $k$ so that $c_{1}q^{{-k}/d}<c^{\prime}/3$.

If $J_{\bf u}\cap J_{\bf v}=\emptyset$, then $K_{\bf u}\cap K_{\bf v}=\emptyset$, it follows that $\hbox{dist}_{w}(K_{\bf u},K_{{\bf v}})\geq c^{\prime}q^{{-n}/d}.$ Applying this to the level $n+k$, we have

[TABLE]

The lemma follows by taking $c=c^{\prime}/3$. ∎

Lemma 3.6**.**

Let $\{S_{i}\}_{i=1}^{N}$ be the IFS as in (1.1). Then for any $b>0$, there exists a constant $\gamma:=\gamma(b)$ such that for any set $V\subset{\mathbb{R}}^{d}$ with $\hbox{diam}_{w}(V)\leq b$,

[TABLE]

where $d_{{\bf u}}=d_{i_{n}}+Ad_{i_{n-1}}+\cdots+A^{n-1}d_{i_{1}}$ for ${\bf u}=i_{1}\cdots i_{n}\in X$.

Proof.

The lemma follows directly from the fact that $\{d_{{\bf u}}:{\bf u}\in X\}\subset{\mathbb{Z}}^{d}$ is uniformly discrete and $J,V$ are bounded subsets of ${\mathbb{R}}^{d}$. ∎

Theorem 3.7**.**

Let $K$ be the integral self-affine set as in (1.1) with $|\det A|=q$. Then the graph $(X,{\mathcal{E}})$ is hyperbolic. Moreover, there exists a Hölder bijection $\phi:\partial X\to K$ satisfying the property:

[TABLE]

where $\alpha=\frac{\log q}{da}$ and $C>0$ is a constant.

Proof.

The proof generalizes the self-similar case by some modifications (see [18] or [24]). For any ${\bf u}\in X$ with $|{\bf u}|=n$, let $V=A^{n}J_{\bf u}$, then $\hbox{diam}_{w}(V)=\hbox{diam}_{w}(J):=b$. By Lemma 3.6, we have

[TABLE]

Hence the graph $(X,{\mathcal{E}})$ is of bounded degree.

Suppose $(X,{\mathcal{E}})$ is not hyperbolic, by Lemma 3.1, then for any $m>0$, there exists a horizontal geodesic $\pi({\bf u}_{0},{\bf u}_{3m})=[{\bf u}_{0},{\bf u}_{1},\dots,{\bf u}_{3m}]$ with ${\bf u}_{i}\in X_{n}$ for some $n$. Consider $m$-th generation ancestors $\{{\bf u}_{0}^{-m},{\bf u}_{1}^{-m},\dots,{\bf u}_{3m}^{-m}\}$. By the definition of the graph, we have either ${\bf u}_{i}^{-m}={\bf u}_{i+1}^{-m}$ or $({\bf u}_{i}^{-m},{\bf u}_{i+1}^{-m})\in{\mathcal{E}}_{h}$. Then there is a path $p({\bf v}_{0},{\bf v}_{1},\dots,{\bf v}_{\ell})$ joining ${\bf u}_{0}^{-m}$ and ${\bf u}_{3m}^{-m}$ where ${\bf v}_{0}={\bf u}_{0}^{-m},{\bf v}_{\ell}={\bf u}_{3m}^{-m}$ and ${\bf v}_{i}\in\{{\bf u}_{0}^{-m},{\bf u}_{1}^{-m},\dots,{\bf u}_{3m}^{-m}\}$. Without loss of generality, we may assume $p({\bf v}_{0},{\bf v}_{1},\dots,{\bf v}_{\ell})$ is the shortest horizontal path joining ${\bf u}_{0}^{-m}$ and ${\bf u}_{3m}^{-m}$. By the geodesic property of $\pi({\bf u}_{0},{\bf u}_{3m})$, it is clear that

[TABLE]

Now choose $m\geq\gamma$ such that $(3m+1)q^{-m/d}\leq 1$, where $\gamma$ is as in Lemma 3.6. Let

[TABLE]

Then

[TABLE]

Note that for each $i$ there exists $j$ such that ${\bf v}_{i}={\bf u}_{j}^{-m}$, it follows that $J_{{\bf u}_{j}}\subset J_{{\bf v}_{i}}$. Thus $J_{{\bf v}_{i}}\cap V^{\prime}\neq\emptyset$. Let $V=A^{n-m}V^{\prime}$. Then $\hbox{diam}_{w}(V)\leq\hbox{diam}_{w}(J)$. It follows that

[TABLE]

which contradicts Lemma 3.6. Therefore, $X$ is hyperbolic.

For any geodesic ray $\xi=\pi[{\mathbf{u}}_{1},{\mathbf{u}}_{2},\ldots]$ of $X$, we define

[TABLE]

for some $x_{0}\in J$. Then the map is well-defined and is bijective (see [24]).

To show that $\phi$ is the desired Hölder map, we let $\xi=\pi[{\mathbf{u}}_{0},{\mathbf{u}}_{1},\ldots],\ \eta=\pi[{\mathbf{v}}_{0},{\mathbf{v}}_{1},\ldots]$ be any two non-equivalent geodesic rays in $X$. Then there is a canonical bilateral geodesic $\tau$ joining $\xi$ and $\eta$:

[TABLE]

with ${\mathbf{u}}_{n},{\mathbf{t}}_{1},\dots,{\mathbf{t}}_{\ell},{\mathbf{v}}_{n}\in X_{n}$. It follows that

[TABLE]

By Lemma 3.1, $\ell$ is uniformly bounded. Note that $\phi(\xi)\in J_{{\mathbf{u}}_{k}}$ and $\phi(\eta)\in J_{{\mathbf{v}}_{k}}$ for all $k\geq 0$, hence

[TABLE]

Using the property $w(x+y)\leq\beta\max\{w(x),w(y)\}$ twice, there exists a constant $C_{1}>0$ such that

[TABLE]

Since $\tau$ is a bilateral canonical geodesic, we have $|\xi\wedge\eta|=n-(\ell+1)/2$ and $\ell$ is uniformly bounded. By using $\rho_{a}(\xi,\eta)=\exp(-a|\xi\wedge\eta|)$, we see that

[TABLE]

On the other hand, assume that $\xi\neq\eta$. Since $\tau$ is a geodesic, it follows that $({\mathbf{u}}_{n+1},{\mathbf{v}}_{n+1})\notin{\mathcal{E}}_{h}$, and hence $J_{{\mathbf{u}}_{n+1}}\cap J_{{\mathbf{v}}_{n+1}}=\emptyset$. By Lemma 3.5, there is $k$ (independent of $n$) such that

[TABLE]

As $\phi(\xi)\in J_{{\bf u}_{n+k+1}},\ \phi(\xi)\in J_{{\bf v}_{n+k+1}}$, we have

[TABLE]

and $w(\phi(\xi)-\phi(\eta))\geq c^{\prime}\rho_{a}(\xi,\eta)^{\alpha}$ follows by the definition of $\rho_{a}$. ∎

4. Lipschitz equivalence of self-affine sets

We first show that the $w$-Hausdorff dimension is an invariant under the $w$-Lipschitz equivalence.

Proposition 4.1**.**

If $E\simeq_{w}F$ then $\dim^{w}_{H}E=\dim^{w}_{H}F$.

Proof.

The proof is the same as Corollary 2.4 of [6] by replacing the Euclidean norm with pseudo-norm $w$. ∎

Let $\lambda_{1}$ and $\lambda_{0}$ be the maximal and minimal moduli of eigenvalues of $A$ defining the pseudo-norm $w$. There is a relationship between $w$-Lipschitz equivalence and nearly Lipschitz equivalence.

Proposition 4.2**.**

Suppose $\lambda_{0}=\lambda_{1}$. If $E\simeq_{w}F$ then $E\simeq_{n}F$.

Proof.

Let $\lambda=\lambda_{0}=\lambda_{1}$ and define a function $h:(0,\lambda-1)\to(0,1)$ by

[TABLE]

Obviously $h$ is a bijection. Hence for any $0<\eta<1$, we can choose $\epsilon\in(0,\lambda-1)$ such that $\eta=h(\epsilon)$.

By taking the bijective map $g(x)=x/|E|$ where $|E|$ is the diameter of $E$ under the Euclidean norm, we have $E\simeq_{n}E/|E|$. Similarly $F\simeq_{n}F/|F|$. Without loss of generality, we may assume $|E|,|F|\leq 1$. Since $E\simeq_{w}F$, there is a bijection $\sigma:E\to F$ satisfying the inequality

[TABLE]

where $C_{0}$ is a constant. That together with Proposition 2.2 implies

[TABLE]

The reverse inequality also follows immediately. By letting $\eta=h(\epsilon)$ as the previous argument, we prove that $E\simeq_{n}F$. ∎

From now on, we focus on the IFS $\{S_{i}\}_{i=1}^{N}$ in (1.1) and fix the invariant set $J=\overline{B_{\delta}}$. Let $J_{k}=\bigcup_{{\bf u}\in\Sigma^{k}}S_{{\bf u}}(J)$ be the $k$-th iteration of $J$ under the IFS, where $S_{\bf u}(J)=A^{-k}(J+d_{\bf u})$. Obviously the self-affine set $K=\bigcap_{k=1}^{\infty}J_{k}$. Denote by

[TABLE]

Then ${\mathbb{H}}=\bigcap_{k=1}^{\infty}{\mathbb{H}}_{k}$.

The following two topological lemmas are straightforward, which were also concerned by Xi and Xiong ([30]).

Lemma 4.3**.**

The union of finitely many totally disconnected compact subsets of ${\mathbb{R}}^{d}$ is also totally disconnected.

Proof.

Let $A_{1},A_{2}$ be totally disconnected compact subsets of ${\mathbb{R}}^{d}$, and let $A=A_{1}\cup A_{2}$. Obviously, if $A_{1}\cap A_{2}=\emptyset$ then $A$ is totally disconnected. Otherwise, we need to show that for any $x\in A$ and any open neighborhood $U$ of $x$ there exists an open-closed set $B$ such that $x\in B\subset U$. Let $x\in A_{1}\cap A_{2}$ and $U$ is an open neighborhood of $x$ in $A$. Then $U\cap A_{i}$ is open in $A_{i}$ for $i=1,2$. Hence there exists an open-closed set $B_{i}$ in $A_{i}$ such that $x\in B_{i}\subset{U\cap A_{i}}$ where $i=1,2$. It follows that $x\in{B_{1}\cup B_{2}}\subset(U\cap A_{1})\cup(U\cap A_{2})=U$. Since $A_{1},A_{2}$ are compact subsets of $A$, we have $B_{1}\cup B_{2}$ is closed in $A$. On the other hand, $U\cap A_{i}$ is also open in $U$ for $i=1,2$. Then $B_{1},B_{2}$ are open in $U$ and $B_{1}\cup B_{2}$ is open in $U$ as well, hence open in $A$. This proves that $B_{1}\cup B_{2}$ is open-closed in $A$. The general case follows by induction. ∎

Lemma 4.4**.**

If the integral self-affine set $K$ is totally disconnected, then there exists $n_{0}\geq 1$ such that any component of $\mathbb{H}_{n_{0}}$ that intersects $\overline{B_{\delta}}$ must lie in $B_{\delta+1}.$

Proof.

For each $n$, let $C_{n}$ be a component of ${\mathbb{H}}_{n}$ that intersects $\overline{B_{\delta}}$. Suppose $C_{n}\cap B_{\delta+1}^{c}\neq\emptyset$. We shall obtain a contraction. Let

[TABLE]

Let $\Gamma_{n}$ be a component of $U_{n}$ that intersects $\overline{B_{\delta}}$. We first show that $\Gamma_{n}$ also intersects the circle $D=\{x\in{\mathbb{R}}^{d}:\|x\|=\delta+1\}$. If not, for any $x\in U_{n}$ with $\|x\|=\delta+1$, there exist two disjoint closed sets $E_{x},F_{x}$ so that $U_{n}=E_{x}\cup F_{x}$ and $x\in F_{x},\Gamma_{n}\subset E_{x}$. By compactness, there is a finite subcover $\{F_{x_{1}},\dots,F_{x_{k}}\}$ of $U_{n}\cap D$. Let

[TABLE]

Then $U_{n}=E\cup F$ with disjoint union. Hence $E$ and $F\cup V_{n}$ form a separation of $C_{n}$, contradicting the assumption of connectedness of $C_{n}$.

Under the Hausdorff metric $d_{H}$, we know that there is a convergent subsequence of $\{\Gamma_{n}\}_{n}$. Without loss of generality, we may assume $\Gamma_{n}\to\Gamma$. Then $\Gamma$ is a connected closed set that intersects both $\overline{B_{\delta}}$ and $D$. Indeed, if $\Gamma$ is not connected, then there is a separation $\Gamma=A\cup B$ where $A,B$ are nonempty closed sets and thus are compact, and

[TABLE]

Let $\Gamma_{n}$ be a component such that $d_{H}(\Gamma_{n},\Gamma)<\epsilon/3.$ Then $\Gamma_{n}$ is contained in an $\epsilon/3$-neighborhood of $A$ and $B$, and $\Gamma_{n}$ cannot be connected. That is ridiculous.

Since $\Gamma_{n}\subset{\mathbb{H}}_{n}\cap\overline{B_{\delta+1}}$ and ${\mathbb{H}}_{n}\cap\overline{B_{\delta+1}}\to{\mathbb{H}}\cap\overline{B_{\delta+1}}$ under the metric $d_{H}$. It follows that $\Gamma\subset{\mathbb{H}}\cap\overline{B_{\delta+1}}$. This contradicts the fact that ${\mathbb{H}}\cap\overline{B_{\delta+1}}$ is totally disconnected by Lemma 4.3. ∎

If a hyperbolic graph $(X,{\mathcal{E}})$ induced by an IFS is of bounded degree, then Theorem 5.5 of [16] shows that $\partial X$* (or the fractal $K$) is totally disconnected if and only if the sizes of horizontal components in $(X,{\mathcal{E}})$ are uniformly bounded.* Under the present setting, the statement can be strengthened as the following version.

Lemma 4.5**.**

The integral self-affine set $K$ is totally disconnected if and only if the graph $(X,{\mathcal{E}})$ is simple.

Proof.

Theorem 3.7 says that $(X,{\mathcal{E}})$ is a hyperbolic graph with bounded degree. If it is simple, then there are only finitely many equivalence classes of horizontal components, hence $K$ is totally disconnected.

Conversely suppose $K$ is totally disconnected. Let $n_{0}$ be a constant in Lemma 4.4. Obviously there are finite equivalence classes of horizontal components in $\bigcup_{j=1}^{n_{0}}X_{j}$. Let $T\subset\bigcup_{j>{n_{0}}}X_{j}$ be a horizontal component. We may assume $T=\{{\bf u}_{1},\dots,{\bf u}_{k}\}\subset X_{n}$ for $n>n_{0}$. Then $J_{T}:=\bigcup_{j=1}^{k}S_{{\bf u}_{j}}(J)$ is connected. Decompose each word ${\bf u}_{j}$ by ${\bf u}_{j}={\bf u}^{1}_{j}{\bf u}^{2}_{j}$ where ${\bf u}^{1}_{j}\in X_{n-n_{0}}$ and ${\bf u}^{2}_{j}\in X_{n_{0}}$. We can write $S_{{\bf u}_{j}}=S_{{\bf u}^{1}_{j}}\circ S_{{\bf u}^{2}_{j}}$. Hence

[TABLE]

Choose ${\bf d}\in{\mathbb{Z}}^{d}$ such that $(A^{n-n_{0}}J_{T}-{\bf d})\cap\overline{B_{\delta}}\neq\emptyset.$ As $A^{n-n_{0}}J_{T}-{\bf d}\subset\mathbb{H}_{n_{0}}$, by Lemma 4.4, we have $A^{n-n_{0}}J_{T}-{\bf d}\subset B_{\delta+1}.$ Hence

[TABLE]

Since $k$ is uniformly bounded and $d_{{\bf u}_{j}}-A^{n_{0}}{\bf d}\in{\mathbb{Z}}^{d}$, there are finitely many connected sets $\bigcup_{j=1}^{k}(J+d_{{\bf u}_{j}})$ in $A^{n_{0}}B_{\delta+1}$, up to translation. Therefore, the equivalence classes of horizontal components of $X$ is finite by definition. ∎

Theorem 4.6**.**

Let $K=A^{-1}(K+{\mathcal{D}}_{1})$ and $K^{\prime}=A^{-1}(K^{\prime}+{\mathcal{D}}_{2})$ be two integral self-affine sets. Suppose both the IFSs satisfy the OSC and $K,K^{\prime}$ are totally disconnected. Then $K\simeq_{w}K^{\prime}$ if and only if $\#{\mathcal{D}}_{1}=\#{\mathcal{D}}_{2}$.

Proof.

If $K\simeq_{w}K^{\prime}$, by Proposition 4.1, then $\dim^{w}_{H}K=\dim^{w}_{H}K^{\prime}$. It follows from Theorem 2.3 that $\#{\mathcal{D}}_{1}=\#{\mathcal{D}}_{2}$.

Conversely, let $\#{\mathcal{D}}_{1}=\#{\mathcal{D}}_{2}=N$ and let $(X,{\mathcal{E}}),(Y,{\mathcal{E}}^{\prime})$ be the hyperbolic graphs induced on $K,K^{\prime}$ respectively. Since the OSC holds, both $(X,{\mathcal{E}}),(Y,{\mathcal{E}}^{\prime})$ are $N$-ary augmented trees satisfying Definition 2.1. From Theorem 3.4 and Lemma 4.5, it yields that

[TABLE]

Let $\varphi:\partial X\to\partial Y$ be a bi-Lipschitz map. By Theorem 3.7, there exist two bijections $\phi_{1}:\partial X\to K$ and $\phi_{2}:\partial Y\to K^{\prime}$ satisfying (3.3) with constants $C_{1},C_{2}$, respectively. Now we define $\sigma:K\to K^{\prime}$ as

[TABLE]

Then

[TABLE]

Let $C^{\prime}=C_{2}C_{0}^{\alpha}C_{1}$, then $w(\sigma(x)-\sigma(y))\leq C^{\prime}w(x-y)$. Moreover, ${C^{\prime}}^{-1}w(x-y)\leq w(\sigma(x)-\sigma(y))$ follows from another inequality of (3.3). Therefore $K\simeq_{w}K^{\prime}$. ∎

Corollary 4.7**.**

Under the assumption of Theorem 4.6. If the eigenvalues of $A$ have equal moduli. Then $K\simeq_{n}K^{\prime}$ if and only if $\#{\mathcal{D}}_{1}=\#{\mathcal{D}}_{2}$.

Proof.

The if part follows from Theorem 4.6 and Proposition 4.2. For the only if part, if $K\simeq_{n}K^{\prime}$, then $\dim_{H}K=\dim_{H}K^{\prime}$. Since the eigenvalues of $A$ have equal moduli, Proposition 2.2 implies that $\dim_{H}^{w}K=\dim_{H}^{w}K^{\prime}$. Therefore, $\#{\mathcal{D}}_{1}=\#{\mathcal{D}}_{2}$ by Theorem 2.3. ∎

If $\mathcal{D}\subset{\mathbb{Z}}^{d}$ is a set of coset representatives of ${\mathbb{Z}}^{d}/{A{\mathbb{Z}}^{d}}$, i.e., $d_{i}+A{\mathbb{Z}}^{d}\neq d_{j}+A{\mathbb{Z}}^{d}$ for distinct $d_{i},d_{j}\in{\mathcal{D}}$. It is well-known that the pair $(A,{\mathcal{D}})$ satisfies the OSC. The following corollary is immediate.

Corollary 4.8**.**

Let $K=A^{-1}(K+{\mathcal{D}}_{1})$ and $K^{\prime}=A^{-1}(K^{\prime}+{\mathcal{D}}_{2})$ be two totally disconnected integral self-affine sets where ${\mathcal{D}}_{1},{\mathcal{D}}_{2}$ are sets of coset representatives of ${\mathbb{Z}}^{d}/{A{\mathbb{Z}}^{d}}$. Then $K\simeq_{w}K^{\prime}$ if and only if $\#{\mathcal{D}}_{1}=\#{\mathcal{D}}_{2}$.

Let A=\left[\begin{array}[]{cc}m&0\\ 0&n\end{array}\right] be an expanding matrix where $m,n\geq 2$ are integers, and let ${\mathcal{D}}\subset\{0,1,\dots,m-1\}\times\{0,1,\dots,n-1\}$ be a digit set. Then the associated self-affine set $K=A^{-1}(K+{\mathcal{D}})$ is the well-known McMullen-Bedford set ([2],[19]). The standard Hausdorff (or box) dimension formula of McMullen-Bedford set has been obtained.

From Corollary 4.8, it can be seen that two totally disconnected McMullen-Bedford sets are $w$-Lipschitz equivalent if and only if their digit sets have equal cardinality. However, the two $w$-Lipschitz equivalent McMullen-Bedford sets may be not Lipschitz equivalent under the Euclidean norm, as they maybe have distinct Hausdorff dimensions.

Example 4.9**.**

Let A=\left[\begin{array}[]{cc}3&0\\ 0&4\end{array}\right] and let three digit sets be as follows

[TABLE]

Let $K,K^{\prime},K^{\prime\prime}$ be the associated McMullen-Bedford sets respectively (see Figure 1). A dimension formula ([19],[6]) yields that

[TABLE]

which are different. Hence $K,K^{\prime\prime}$ and $K^{\prime},K^{\prime\prime}$ are both not Lipschitz equivalent under the Euclidean norm. It is also not clear that if $K\simeq K^{\prime}$ even $\dim_{H}K=\dim_{H}K^{\prime}$.

On the other hand, by using a criterion for integral self-affine sets to be totally disconnected ([31]), it can be verified that $K,K^{\prime},K^{\prime\prime}$ are all totally disconnected. Therefore, $K\simeq_{w}K^{\prime}\simeq_{w}K^{\prime\prime}$ by Corollary 4.8. **

Acknowledgements: The author gratefully acknowledges the support of K. C. Wong Education Foundation and DAAD. He also would like to thank Prof. Martina Zähle for valuable discussions.