Graph-Theoretic Characterization of Separation Conditions in Self-Affine Iterated Function Systems

Abstract

For a self-affine iterated function system (IFS) ${\left\{f_j\right\}}_{j=1}^N$ on ${\mathbb{R}}^d$ defined by $f_j\left(x\right)=A^{-1}(x+d_j)$, where A is an expansive matrix and $d_j\in {\mathbb{R}}^d$, we reveal a novel characterization of the open set and weak separation conditions through the bounded degree of the augmented tree induced by the IFS. Furthermore, the augmented tree is shown to be a Gromov hyperbolic graph, and its hyperbolic boundary is Hölder equivalent to the self-affine set generated by the IFS, establishing a canonical association between self-affine IFSs and Gromov hyperbolic graphs.

1. Introduction

Let A be a $d\times d$ matrix with real entries that is expansive, i.e., all the eigenvalues of A have moduli $>1$. Given an integer $N>1$, let $\mathcal{D}=\{d_1,d_2,\dots ,d_N\}\subset {\mathbb{R}}^d$ be a set of N vectors. Define affine contracting functions on ${\mathbb{R}}^d$ as follows:

$$f_j\left(x\right)=A^{-1}(x+d_j),j=1,\dots ,N.$$

(1)

The family of functions ${\left\{f_j\right\}}_{j=1}^N$ is called a self-affine iterated function system (IFS) [1]. According to Hutchinson’s theorem [2], there exists a unique, nonempty, compact set $K:=K(A,\mathcal{D})$ that satisfies the following:

$$K=\bigcup _{j=1}^N{f}_j\left(K\right)=A^{-1}(K+\mathcal{D}).$$

(2)

The compact set K is canonically termed a self-affine set or an attractor of the IFS. When $N=|det(A\left)\right|$ holds and K possesses a nonempty interior, the attractor is classified as a self-affine tile [3], characterized by its capacity to tessellate the ambient space through translational copies. In a specialized case where the linear component A constitutes a similitude satisfying $A^{-1}=rR$ with $0<r<1$ (scaling factor) and an orthonormal matrix R, the system ${\left\{f_j\right\}}_{j=1}^N$ is designated as a self-similar IFS, and its attractor K accordingly becomes a self-similar set.

Both the IFS and its attractor are the main research objects of fractal geometry and dynamical systems. Let ${\Sigma }^{*}:={\bigcup }_{n=0}^{\infty }{\Sigma }^n$ be the symbolic space associated with the IFS, where $\Sigma =\{1,\dots ,N\}$ and ${\Sigma }^0=\varnothing$. Then, write the following:

$${\mathcal{D}}_1=\mathcal{D},{\mathcal{D}}_n=\mathcal{D}+A{\mathcal{D}}_{n-1},n\ge 2,\mathrm{and}{\mathcal{D}}_{\infty }=\bigcup _{n=1}^{\infty }{\mathcal{D}}_n.$$

If $\mathbf{u}=i_1\cdots i_n\in {\Sigma }^{*}$, we assume $f_{\mathbf{u}}=f_{i_1}\circ \cdots \circ f_{i_n}$ for the sake of simplicity. Then, we consider $f_{\mathbf{u}}\left(x\right)=A^{-n}(x+d_{\mathbf{u}})$ with the following:

$$d_{\mathbf{u}}=d_{i_n}+A{d}_{i_{n-1}}+\cdots +A^{n-1}{d}_{i_1}\in {\mathcal{D}}_n.$$

Next, write $K_{\mathbf{u}}:=f_{\mathbf{u}}\left(K\right)=A^{-n}(K+d_{\mathbf{u}})$. By iterating (2), the self-affine set K is of the following form for any $n\ge 1$:

$$K=\bigcup _{\mathbf{u}\in {\Sigma }^n}{f}_{\mathbf{u}}\left(K\right)=A^{-n}(K+{\mathcal{D}}_n).$$

The symbolic space ${\Sigma }^{*}$ naturally inherits a tree structure through standard word concatenation, where we denote the vertical edge set by ${\mathcal{E}}_v$. This construction explicitly reveals $({\Sigma }^{*},{\mathcal{E}}_v)$ as an N-ary tree. Such a combinatorial structure plays a crucial role in calculating the Hausdorff dimension of the associated attractor K (see [1,2,4]).

In the current field of fractal geometry, the open set condition is perhaps one of the most fundamental and useful separation conditions to track the iteration rules. The IFS ${\left\{f_j\right\}}_{j=1}^N$ is said to satisfy the open set condition (OSC) [1,2], if there is a nonempty, bounded, open set $U\subset {\mathbb{R}}^d$ such that ${\bigcup }_{j=1}^N{f}_j\left(U\right)\subset U$ and $f_i\left(U\right)\cap f_j\left(U\right)=\varnothing$ for all $i\ne j$. If the additional condition $U\cap K\ne \varnothing$ holds, we say that the IFS satisfies the strong open set condition (SOSC).

Self-similar IFSs with the OSC can be handled easily, and the Hausdorff dimension of self-similar sets can be calculated accurately. In particular, Schief [4] proved a well-known result regarding the characterization of the OSC in self-similar IFSs. Recently, Fu, Gabardo, and Qiu [5] extended Schief’s result and obtained a generalization on self-affine IFSs by replacing the Euclidean metric with a quasi-metric $d_w$ introduced by He and Lau [6]. Under the quasi-metric $d_w$ (see details in Section 2), the self-affine IFS (1) exhibited a similarity structure as

$$d_w(A^{-1}x,A^{-1}y)=q^{-{\displaystyle \frac{1}{d}}}{d}_w(x,y),x,y\in {\mathbb{R}}^d,$$

where $q=|detA|>1$. It allowed for better dimensional estimates of K through the generalized Hausdorff measure ${\mathcal{H}}_w^s$ and the generalized Hausdorff dimension ${dim}_H^w$ (please refer to [6] for definitions).

Theorem 1

([5,6]). Let the IFS ${\left\{f_j\right\}}_{j=1}^N$ be as in (1), then the following are equivalent:

(i) 

the OSC holds;

(ii) 

the SOSC holds;

(iii) 

${\mathcal{H}}_w^s\left(K\right)>0$, where $s=dlogN/logq$;

(iv) 

$\#{\mathcal{D}}_n=N^n$ for all $n\ge 1$ and ${\mathcal{D}}_{\infty }$ is uniformly discrete, i.e. there exists $\delta >0$ such that $\parallel d_{\mathbf{u}}-d_{\mathbf{v}}\parallel >\delta$ for any $d_{\mathbf{u}}\ne d_{\mathbf{v}}\in {\mathcal{D}}_{\infty }$.

When the OSC fails for an IFS, iterations inevitably generate overlaps, and the resulting attractor K exhibits intricate geometric complexity. To address this challenge, the weak separation condition (WSC) introduced by Lau and Ngai [7] (see also [8]) serves as a pivotal tool for analyzing self-similar IFSs with overlaps. While weaker than the OSC, the WSC retains key analytical properties (e.g., controlled overlapping scales) and maintains the tractability of the iterative process.

Deng [9] generalized the concept of WSC as follows: A self-affine IFS ${\left\{f_j\right\}}_{j=1}^N$ satisfies the WSC if, for every bounded set $D\subset {\mathbb{R}}^d$, there exists $\gamma >0$ such that

$$\#\left\{f_{\mathbf{u}}:\mathbf{u}\in {\Sigma }^n,x\in f_{\mathbf{u}}\left(D\right)\right\}<\gamma \mathrm{for}\mathrm{all}x\in {\mathbb{R}}^d\mathrm{and}n\ge 1.$$

This condition effectively quantifies overlapping multiplicity while accommodating affine distortions. A systematic exposition of the OSC, WSC, and other separation conditions and their relationships was comprehensively presented in the survey paper [10].

Theorem 2

([9]). Let the IFS ${\left\{f_j\right\}}_{j=1}^N$ be as in (1), then the following are equivalent:

(i) 

the WSC holds;

(ii) 

there exists $\delta >0$ such that for any $d_{\mathbf{u}},d_{\mathbf{v}}\in {\mathcal{D}}_n,n\ge 1$, either $d_{\mathbf{u}}=d_{\mathbf{v}}$ or $\parallel d_{\mathbf{u}}-d_{\mathbf{v}}\parallel >\delta$;

(iii) 

for any given bounded subsets $D_1,D_2\subset {\mathbb{R}}^d$, there exists ${\gamma }^{\prime }>0$ such that $\#\{d_{\mathbf{u}}:\mathbf{u}\in {\Sigma }^n,(D_1+x)\cap (D_2+d_{\mathbf{u}})\ne \varnothing \}<{\gamma }^{\prime }$ for all $x\in {\mathbb{R}}^d$ and $n\ge 1$;

(iv) 

for any $c>0$, there exists a constant $\beta \left(c\right)$ such that for any integer $n\ge 1$ and any bounded set $D\subset {\mathbb{R}}^d$ with diameter $\le c{r}^n$,

$$\#\{f_{\mathbf{u}}:\mathbf{u}\in {\Sigma }^n,x\in f_{\mathbf{u}}\left(D\right)\}<\beta \left(c\right).$$

(3)

Motivated by the previous works, we propose a novel characterization of the OSC and WSC in a self-affine IFS through graph-theoretic invariants. Before presenting our main theorems, we need some notation on graphs.

For an N-ary tree $({\Sigma }^{*},{\mathcal{E}}_v)$, we define horizontal edges as follows:

$${\mathcal{E}}_h=\{(\mathbf{u},\mathbf{v})\in {\Sigma }^n\times {\Sigma }^n:\mathbf{u}\ne \mathbf{v}, d_w(K_{\mathbf{u}},K_{\mathbf{v}})\le \rho r^n, n\ge 1\}$$

where $d_w(E,F)=inf\{d_w(x,y):x\in E,y\in F\},r=q^{-\frac{1}{d}}$, and $\rho >0$ is a constant. Let $\mathcal{E}={\mathcal{E}}_v\cup {\mathcal{E}}_h$. Then, the graph $G:=({\Sigma }^{*},\mathcal{E})$ is an augmented tree (see Definition 2). The concept of augmented tree was initially introduced by Kaimanovich [11] to study the Sierpinski graph. Later, Lau and Wang [12] developed his idea into a large class of self-similar sets.

If the IFS in (1) satisfies the OSC, by Theorem 1, $d_{\mathbf{u}}\ne d_{\mathbf{v}}$ holds for any distinct $\mathbf{u},\mathbf{v}\in {\Sigma }^{*}$. If the IFS does not satisfy the OSC, it may occur that $d_{\mathbf{u}}=d_{\mathbf{v}}$ for $\mathbf{u}\ne \mathbf{v}\in {\Sigma }^{*}$. In this case, we may modify the augmented tree $({\Sigma }^{*},\mathcal{E})$ by identifying $\mathbf{u},\mathbf{v}$ for $d_{\mathbf{u}}=d_{\mathbf{v}}$, and let $G^{\sim }$ be the quotient space of the graph $G=({\Sigma }^{*},\mathcal{E})$. The detailed modification can be found in Section 3.

A graph $(X,\mathcal{E})$ is of bounded degree if $sup\{deg(x):x\in X\}<\infty$, where $deg\left(x\right)=\#\{y:(x,y)\in \mathcal{E}\}$ is the total number of edges joining x.

Theorem 3.

Let the IFS be as in (1). Then, we have the following:

(i) 

The OSC holds if and only if the graph G is of bounded degree.

(ii) 

The WSC holds if and only if the quotient space $G^{\sim }$ is of bounded degree.

The idea of the proof is based on graph theory and the iteration rule of an IFS under the quasi-metric $d_w$. We provide an example (see Example 1) to illustrate the conclusion.

Recently, Gromov hyperbolicity andthe hyperbolic boundaries of augmented trees induced by self-similar IFSs have attracted considerable attention (see [11,12,13,14,15,16,17]). However, there are very few related studies on self-affine IFSs. In the second part of this paper, we demonstrate that the augmented tree G inherently possesses Gromov hyperbolicity. This enables the establishment of a canonical association between self-affine IFSs and Gromov hyperbolic graphs, notably achieved without imposing separation conditions.

Theorem 4.

For any IFS as in (1), the augmented tree $G:=({\Sigma }^{*},\mathcal{E})$ is always a Gromov hyperbolic graph (see Definition 1).

Moreover, the hyperbolic boundary $\partial G$ is Hölder equivalent to the self-affine set K, that is, there exists a homeomorphism $\psi :\partial G\to K$ such that the following can be obtained::

$$C^{-1}{d}_w\left(\psi \left(\eta \right),\psi \left(\xi \right)\right)\le {\theta }_{\tau }^{\alpha }(\eta ,\xi )\le C{d}_w\left(\psi \left(\eta \right),\psi \left(\xi \right)\right),\forall \eta ,\xi \in \partial G,$$

(4)

where $C>0,\alpha ={\displaystyle \frac{logq}{d\tau }},q=|detA|$ and ${\theta }_{\tau }$ is a visual metric on G.

The paper is organized as follows: Section 2 describes the basic definitions and properties of Gromov hyperbolic graphs and augmented trees induced by IFSs. Section 3 is devoted to establishing Theorem 3 (by proving Theorems 5 and 6) and Theorem 4.

2. Augmented Tree Induced by IFS

Following the notation in graph theory, let $(X,\mathcal{E})$ be a connected graph, where X is a countably infinite set of vertices and $\mathcal{E}\subset X\times X$ stands for the set of edges. Let $\pi (x,y)$ denote a geodesic path from x to y in X, i.e., a path connecting $x,y$ with the shortest length. Denote the distance between x and y as $d(x,y)$. The degree of a vertex x is defined by $deg\left(x\right)=\#\{y:(x,y)\in \mathcal{E}\}$, the total number of edges connecting x. A graph $(X,\mathcal{E})$ is of a bounded degree if $sup\{deg(x):x\in X\}<\infty$.

The bounded degree of a graph is an important tool for studying the random walks on graphs; however, it should be distinguished from another concept, “locally finite”. A graph is said to be locally finite if every vertex has finite degree (i.e., $deg\left(x\right)<\infty$).

Fix a vertex o in X as a root of the graph. Let $\left|x\right|=d(o,x)$ denote the distance from the root o to the vertex x. $\left|x\right|=n$ means that x is located on the n-th level of the graph. A graph is called a tree if any pair of vertices can be connected by a unique path.

Definition 1

([18]). A graph $(X,\mathcal{E})$ is called a Gromov hyperbolic graph, if there exists $\delta \ge 0$ satisfying the following for all $x,y,z\in X$:

$$|x\wedge y|\ge min\left\{\right|x\wedge z|,|z\wedge y\left|\right\}-\delta$$

where $|x\wedge y|$ is the Gromov product of x and y defined by the following:

$$|x\wedge y|:={\displaystyle \frac{1}{2}}\left(\right|x|+|y|-d(x,y)).$$

If $(X,\mathcal{E})$ is a Gromov hyperbolic graph, we may choose a number $\tau >0$ such that $exp\left(3\delta \tau \right)<\sqrt{2}$. Define a function on $X\times X$ as follows:

$${\theta }_{\tau }(x,y)=\left\{\begin{array}{cc}exp(-\tau |x\wedge y\left|\right)\hfill & x\ne y\hfill \\ 0\hfill & x=y.\hfill \end{array}\right.$$

The function ${\theta }_{\tau }$ is equivalent to a metric [19]. Hence, it can be treated as a visual metric for convenience. Let $\overline{X}$ be the ${\theta }_{\tau }$-completion of X. We call $\partial X=\overline{X}\setminus X$ the hyperbolic boundary of the Gromov hyperbolic graph X.

Let X be a tree with a root o. It can be easily seen that X is a Gromov hyperbolic graph 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. Let $x^{-1}$ be the father of x, i.e., if $x\in X_n$ for $n\ge 1$, then there exists a unique $x^{-1}\in X_{n-1}$ such that $(x^{-1},x)\in {\mathcal{E}}_v$. Inductively, we can define $x^{-m}={\left(x^{-(m-1)}\right)}^{-1}$ as the m-th generation ancestor of x. If every vertex has N offspring, we would call $(X,{\mathcal{E}}_v)$ an N-ary tree.

Definition 2

([11]). Let $(X,{\mathcal{E}}_v)$ be a tree with a root o, and let ${\mathcal{E}}_h\subset (X\times X)\setminus \{(x,x):x\in X\}$ be symmetric. If ${\mathcal{E}}_h$ satisfies the following:

$$(x,y)\in {\mathcal{E}}_h\Rightarrow \left|x\right|=\left|y\right|,andeither{x}^{-1}=y^{-1}or(x^{-1},y^{-1})\in {\mathcal{E}}_h,$$

where $\mathcal{E}={\mathcal{E}}_v\cup {\mathcal{E}}_h$, we would call $(X,\mathcal{E})$ an augmented tree. We call elements in ${\mathcal{E}}_h$ horizontal edges.

Moreover, if $(X,{\mathcal{E}}_v)$ is an N-ary tree, we would call $(X,\mathcal{E})$ an N-ary augmented tree.

In general, the geodesic paths from x to y in an augmented tree $(X,\mathcal{E})$ are not unique. However, there always exists a canonical geodesic of the following form:

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

where $\pi (x,u),\pi (v,y)$ are vertical paths; and $\pi (u,v)$ is a horizontal path (see Figure 1). One or two parts may vanish in the canonical form. For any geodesic ${\pi }^{\prime }(x,y)$, the distance from the root o to the geodesics satisfy the following: $d(o,\pi (u,v))\le d(o,{\pi }^{\prime }(x,y))$. Trivially, the Gromov product of $x,y$ can be calculated as follows:

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

(5)

where $h,l$ are the level and length of the horizontal part of the canonical geodesic $\pi (x,y)$, respectively.

Lemma 1

([12]). An augmented tree is a Gromov hyperbolic graph if and only if there exists $\kappa >0$ such that the length of any horizontal geodesic is bounded by κ.

A sequence ${\left\{x_n\right\}}_{n=0}^{\infty }\subset X$ is called a geodesic ray and denoted by $\pi (x_0,x_1,\dots )$, if $x_0=o,\left|x_n\right|=n$ and $(x_n,x_{n+1})\in {\mathcal{E}}_v$. It is useful to identify $\eta \in \partial X$ with equivalent rays that converge to $\eta$. Two geodesic rays $\pi (x_0,x_1,\dots )$ and $\pi (y_0,y_1,\dots )$ are called equivalent [19] if and only if there is $L>0$ such that

$$d(x_n,y_n)\le L$$

for all but finitely many n. Moreover, the Gromov product can be extended to the boundary $\partial X$ by assuming the following:

$$|\eta \wedge \xi |=inf\{\underset{n\to \infty }{lim}|x_n\wedge y_n|\}$$

where $\eta ,\xi \in \partial X$; and the infimum has taken over all geodesic rays $\pi (x_0,x_1,\dots )$ and $\pi (y_0,y_1,\dots )$, converging to $\eta$ and $\xi$, respectively.

Given an expansive $d\times d$ matrix A with $|detA|=q>1$, the matrix A can define a pseudo-norm w on ${\mathbb{R}}^d$, which satisfies the following fundamental properties:

Proposition 1

([6]). (i) $w\left(x\right)=w(-x)$ and $w\left(x\right)\ge 0$, where $w\left(x\right)=0$ if and only if $x=0$;

(ii) 

$w\left(Ax\right)=q^{{\displaystyle \frac{1}{d}}}w\left(x\right)\ge w\left(x\right),\forall x\in {\mathbb{R}}^d$;

(iii) 

there is $C>0$, such that for any $x,y\in {\mathbb{R}}^d$, $w(x+y)\le Cmax\left\{w\right(x),w(y\left)\right\}$.

The w induces a w-distance, as follows:

$$d_w(x,y)=w(x-y),x,y\in {\mathbb{R}}^d.$$

It is easy to verify that $({\mathbb{R}}^d,d_w)$ is a complete quasi-metric space [20]. Moreover, we can define the w-ball $B_w(z,r)=\{x:d_w(z,x)\le r\}$; the w-diameter of a set ${\mathrm{diam}}_w\left(E\right)=sup\{d_w(x,y):x,y\in E\}$; and the w-distance between two sets $d_w(E,F)=inf\{d_w(x,y):x\in E,y\in F\}$.

Let the IFS ${\left\{f_j\right\}}_{j=1}^N$ be as in (1); Proposition 1 (ii) tells us the following:

$$d_w(f_j\left(x\right),f_j\left(y\right))=q^{-{\displaystyle \frac{1}{d}}}{d}_w(x,y),x,y\in {\mathbb{R}}^d.$$

Hence, every $f_j$ can be regarded as a contracting similitude with contraction ratio $r:=q^{-{\displaystyle \frac{1}{d}}}$ in the quasi-metric space $({\mathbb{R}}^d,d_w)$. Let $\Sigma =\{1,\dots ,N\}$ and ${\Sigma }^{*}:={\bigcup }_{n=0}^{\infty }{\Sigma }^n$ be the symbolic space, where ${\Sigma }^0=\varnothing$. There is a natural tree structure on ${\Sigma }^{*}$ by the standard concatenation of words, and we denote the edge set by ${\mathcal{E}}_v$. That is, $(\mathbf{u},\mathbf{v})\in {\mathcal{E}}_v$ if and only if there exists $\ell \in \Sigma$ such that $\mathbf{u}=\mathbf{v}\ell$ or $\mathbf{v}=\mathbf{u}\ell$.

Let K be the self-affine set as in (2), and let $K_{\mathbf{u}}:=f_{\mathbf{u}}\left(K\right),\mathbf{u}\in {\Sigma }^{*}$ be as in the introduction. Then, we define a set of horizontal edges on ${\Sigma }^{*}$ as follows:

$${\mathcal{E}}_h=\{(\mathbf{u},\mathbf{v})\in {\Sigma }^n\times {\Sigma }^n:\mathbf{u}\ne \mathbf{v}, d_w(K_{\mathbf{u}},K_{\mathbf{v}})\le \rho r^n,n\ge 1\}$$

(6)

where $r=q^{-{\displaystyle \frac{1}{d}}}<1$ is the contraction ratio of the IFS under quasi-metric $d_w$; and $\rho >0$ is a constant.

Let $\mathcal{E}={\mathcal{E}}_v\cup {\mathcal{E}}_h$. It follows from Definition 2 that the graph $G:=({\Sigma }^{*},\mathcal{E})$ is an augmented tree induced by the IFS ${\left\{f_j\right\}}_{j=1}^N$ in the sense of Kaimanovich.

Lemma 2.

Let ${\left\{f_j\right\}}_{j=1}^N$ the IFS as in (1). Suppose that the induced augmented tree $(X,\mathcal{E})$ is of bounded degree, then $f_{\mathbf{u}}\ne f_{\mathbf{v}}$ for any $\mathbf{u}\ne \mathbf{v}\in {\Sigma }^{*}$.

Proof. 

If otherwise, there are $\mathbf{u}\ne \mathbf{v}\in {\Sigma }^{*}$ such that $f_{\mathbf{u}}=f_{\mathbf{v}}$. For any $k\ge 1$, denote as follows:

$${\mathcal{G}}_k=\{\mathbf{x}={\mathbf{x}}_1\cdots {\mathbf{x}}_k:{\mathbf{x}}_i=\mathbf{u}\mathrm{or}\mathbf{v},i=1,\dots ,k\}.$$

Then, $f_{\mathbf{x}}=f_{\mathbf{y}}$ for all $\mathbf{x},\mathbf{y}\in {\mathcal{G}}_k$. Hence, $(\mathbf{x},\mathbf{y})\in {\mathcal{E}}_h$ for all $\mathbf{x},\mathbf{y}\in {\mathcal{G}}_k$, yielding that

$$deg\left(\mathbf{x}\right)\ge 2^k-1,\mathrm{for}\mathrm{all}\mathbf{x}\in {\mathcal{G}}_k.$$

This contradicts the assumption that $({\Sigma }^{*},\mathcal{E})$ is of bounded degree. □

Lemma 3

([20]). If the IFS in (1) satisfies the OSC, then we obtain the following:

(*) for any $c>0$, there exists $\beta \left(c\right)$ such that for any $n\ge 1$, and any w-ball D with radius $c{r}^n$ in ${\mathbb{R}}^d$, we have $\#\{\mathbf{u}\in {\Sigma }^n:K_{\mathbf{u}}\cap D\ne \varnothing \}\le \beta \left(c\right).$

3. Proof of Main Results

Theorem 5.

The IFS in (1) satisfies the OSC if and only if the graph $G:=({\Sigma }^{*},\mathcal{E})$ is of bounded degree.

Proof. 

If the OSC holds, for any $\mathbf{u}\in {\Sigma }^{*}$ with $\left|\mathbf{u}\right|=n$, then $\mathbf{u}\in {\Sigma }^n$ and ${\mathrm{diam}}_w\left(K_{\mathbf{u}}\right)=r^n{\mathrm{diam}}_w\left(K\right)$, where $r=q^{-{\displaystyle \frac{1}{d}}}$. Let D be a w-ball, with the center at a point in $K_{\mathbf{u}}$ and the radius $({\mathrm{diam}}_w\left(K\right)+2\rho )r^n$. For any horizontal edge $(\mathbf{u},\mathbf{v})\in {\mathcal{E}}_h$, we have $\mathbf{v}\in {\Sigma }^n$ and $d_w(K_{\mathbf{u}},K_{\mathbf{v}})\le \rho r^n$. Hence, $K_{\mathbf{v}}\cap D\ne \varnothing$. It follows from Lemma 3 that

$$\#\{\mathbf{v}\in {\Sigma }^n:(\mathbf{u},\mathbf{v})\in {\mathcal{E}}_h\}\le \beta (2\rho +{\mathrm{diam}}_w\left(K\right)).$$

Since $({\Sigma }^{*},{\mathcal{E}}_v)$ is an N-ary tree, there are at most $N+1$ terms $\mathbf{v}$ such that $(\mathbf{u},\mathbf{v})\in {\mathcal{E}}_v$ is a vertical edge. Thus, the degree of $\mathbf{u}$ is as follows:

$$deg\left(\mathbf{u}\right)\le N+1+\beta (2\rho +{\mathrm{diam}}_w\left(K\right)),$$

proving that $(X,\mathcal{E})$ is of bounded degree.

On the contrary, first of all, we claim that the condition (*) in Lemma 3 is true. Otherwise, there is $c>0$; for any $\beta >0$, there is $n\ge 1$ and a w-ball D with radius $c{r}^n$, as follows:

$$\#\{\mathbf{u}\in {\Sigma }^n:K_{\mathbf{u}}\cap D\ne \varnothing \}>\beta .$$

Let $X_{n,D}=\{\mathbf{u}\in {\Sigma }^n:K_{\mathbf{u}}\cap D\ne \varnothing \}$ and let

$$D^{\prime }=\bigcup _{\mathbf{u}\in X_{n,D}}{K}_{\mathbf{u}}.$$

It follows that

$${\mathrm{diam}}_w\left(D^{\prime }\right)\le 2{\mathrm{diam}}_w\left(K\right)r^n+2c{r}^n=2({\mathrm{diam}}_w\left(K\right)+c)r^n.$$

Select a family of open ball sets $\{D_1,\dots ,D_{k_0}\}$ with radius ${\displaystyle \frac{1}{2}}\rho r^n$ to cover $D^{\prime }$, where $k_0$ is independent of n and $\rho$ is a constant in the definition of ${\mathcal{E}}_h$. According to the pigeonhole principle, there exists a ball $D_{i_0}$ that intersects at least ${\beta }^{\prime }:=[{\displaystyle \frac{\beta }{k_0}}]$ (the largest integer $\le {\displaystyle \frac{\beta }{k_0}}$) sets of $K_{\mathbf{u}},\mathbf{u}\in X_{n,D}$, say, $K_{{\mathbf{u}}_1},K_{{\mathbf{u}}_2},\dots ,K_{{\mathbf{u}}_{{\beta }^{\prime }}}$. This implies that, for any integers $i,j\in [1,{\beta }^{\prime }]$, $d_w(K_{{\mathbf{u}}_i},K_{{\mathbf{u}}_j})\le \rho r^n$. By (6), $({\mathbf{u}}_i,{\mathbf{u}}_j)\in {\mathcal{E}}_h(i\ne j)$, hence

$$deg\left({\mathbf{u}}_i\right)\ge {\beta }^{\prime }-1i=1,\dots ,{\beta }^{\prime }.$$

Since $\beta$ can be sufficiently large, ${\beta }^{\prime }$ can also be sufficiently large. It contradicts the assumption of bounded degree. We have proven the claim.

Finally, we show that the OSC is satisfied by constructing the desired open set. For the IFS in (1), take an open ball $D\subset {\mathbb{R}}^d$ satisfying $K\subset {\bigcup }_{j=1}^N{f}_j\left(D\right)\subset D$. The above claim allows us to yield the following:

$$\gamma =\underset{n\ge 1}{sup}\underset{\mathbf{u}\in {\Sigma }^n}{max}\#\{f_{\mathbf{v}}:\mathbf{v}\in {\Sigma }^n,f_{\mathbf{u}}\left(D\right)\cap f_{\mathbf{v}}\left(D\right)\ne \varnothing \}<\infty .$$

Hence, there are ${\mathbf{u}}_1,\dots ,{\mathbf{u}}_{\gamma }\in {\Sigma }^n$ for some $n\ge 1$ such that $f_{{\mathbf{u}}_1}\left(D\right)\cap f_{{\mathbf{u}}_i}\left(D\right)\ne \varnothing$ and $f_{{\mathbf{u}}_i}$’s are distinct. Let

$$U=\bigcup _{\mathbf{v}\in {\Sigma }^{*}}{f}_{\mathbf{v}{\mathbf{u}}_1}\left(D\right).$$

It suffices to show that U is the open set defining the OSC. Clearly, $U\subset D$ is open and bounded and ${\cup }_{j=1}^N{f}_j\left(U\right)\subset U$. If the union is not disjointed, then there exists two distinct $i_0,j_0$ such that $f_{i_0}\left(U\right)\cap f_{j_0}\left(U\right)\ne \varnothing$. From the definition of U, it follows that there are ${\mathbf{v}}_1,{\mathbf{v}}_2\in {\Sigma }^{*}$, which can be expressed as follows:

$$f_{i_0{\mathbf{v}}_1{\mathbf{u}}_1}\left(D\right)\cap f_{j_0{\mathbf{v}}_2{\mathbf{u}}_1}\left(D\right)\ne \varnothing .$$

Without loss of generality, we may assume that the two words $i_0{\mathbf{v}}_1{\mathbf{u}}_1,j_0{\mathbf{v}}_2{\mathbf{u}}_1$ lie in the same level, say ${\Sigma }^{n_1}$ (otherwise, we can choose a prefix of the longer one).

$$\{f_{\mathbf{v}}:\mathbf{v}\in {\Sigma }^{n_1},f_{i_0{\mathbf{v}}_1{\mathbf{u}}_1}\left(D\right)\cap f_{\mathbf{v}}\left(D\right)\ne \varnothing \}\supset \{f_{i_0{\mathbf{v}}_1{\mathbf{u}}_i}:i=1,\dots ,\gamma \}\cup \left\{f_{j_0{\mathbf{v}}_2{\mathbf{u}}_1}\right\}.$$

It follows from Lemma 2 that the right-hand side set contains $\gamma +1$ different functions, which contradicts the maximality of $\gamma$. We have completed the proof. □

In the absence of the OSC, we need to modify the graph $G=({\Sigma }^{*},\mathcal{E})$ as follows. First, we define a relation ∼ on the symbolic space ${\Sigma }^{*}$. For $\mathbf{u},\mathbf{v}\in {\Sigma }^{*}$, denote by $\mathbf{u}\sim \mathbf{v}$ if $f_{\mathbf{u}}=f_{\mathbf{v}}$. Trivially, ∼ is an equivalence relation. Let ${\Sigma }^{*}/\sim$ be the quotient of ${\Sigma }^{*}$ and $\left[\mathbf{u}\right]$ the equivalence class containing $\mathbf{u}$. In this situation, $(\mathbf{u},\mathbf{v})\in {\mathcal{E}}_v$ if and only if there exist $\mathbf{i}\in \left[\mathbf{u}\right],\mathbf{j}\in \left[\mathbf{v}\right]$ and some $\ell \in \Sigma$, such that $\mathbf{j}=\mathbf{i}\ell$ or $\mathbf{i}=\mathbf{j}\ell$. We call $G^{\sim }:=({\Sigma }^{*}/\sim ,\mathcal{E})$ the quotient space of the graph $G=({\Sigma }^{*},\mathcal{E})$. Subsequently, we still consider $\mathbf{u}$ as $\left[\mathbf{u}\right]$ for convenience. It should be mentioned that we have $G^{\sim }=G$ if the OSC holds.

Theorem 6.

The IFS in (1) satisfies the WSC if and only if the quotient space $G^{\sim }:=({\Sigma }^{*}/\sim ,\mathcal{E})$ is of bounded degree.

Proof. 

The proof of sufficiency is analogous to that of Theorem 5, so we only need to show the necessity. For any $\left[\mathbf{u}\right]\in {\Sigma }^{*}/\sim$ with $\left|\mathbf{u}\right|=n$, then $\mathbf{u}\in {\Sigma }^n$. Let $D_{\mathbf{u}}$ be the $\left(\rho r^n\right)$-neighborhood of $K_{\mathbf{u}}$, as follows:

$$D_{\mathbf{u}}=\{x\in {\mathbb{R}}^d:d_w(x,K_{\mathbf{u}})\le \rho r^n\}.$$

By the definition of horizontal edge set, for any $(\mathbf{u},\mathbf{v})\in {\mathcal{E}}_h$, we have $d_w(K_{\mathbf{u}},K_{\mathbf{v}})\le \rho r^n$, that means $K_{\mathbf{v}}\cap D_{\mathbf{u}}\ne \varnothing$. It is easy to verify the following:

$${\mathrm{diam}}_w\left(D_{\mathbf{u}}\right)\le {\mathrm{diam}}_w\left(K_{\mathbf{u}}\right)+2\rho r^n=({\mathrm{diam}}_w\left(K\right)+2\rho )r^n.$$

By (3), we have the following:

$$\#\{\mathbf{v}\in {\Sigma }^{*}/\sim :(\mathbf{u},\mathbf{v})\in {\mathcal{E}}_h\}\le \beta ({\mathrm{diam}}_w\left(K\right)+2\rho ).$$

For the vertical edges $(\mathbf{u},\mathbf{v})\in {\mathcal{E}}_v$, we have $\left|\mathbf{v}\right|=\left|\mathbf{u}\right|+1$ or $\left|\mathbf{v}\right|=\left|\mathbf{u}\right|-1$, and $\varnothing \ne K_{\mathbf{u}}\cap K_{\mathbf{v}}\subset D_{\mathbf{u}}\cap K_{\mathbf{v}}$. By using (3) again, we have the following:

$$\#\{\mathbf{v}\in {\Sigma }^{*}/\sim :(\mathbf{u},\mathbf{v})\in {\mathcal{E}}_v\}\le \beta \left(r({\mathrm{diam}}_w\left(K\right)+2\rho )\right)+\beta \left(r^{-1}({\mathrm{diam}}_w\left(K\right)+2\rho )\right).$$

Therefore, combining the horizontal and vertical edges, we can draw the following conclusion:

$$deg\left(\mathbf{u}\right)\le \beta ({\mathrm{diam}}_w\left(K\right)+2\rho )+\beta \left(r({\mathrm{diam}}_w\left(K\right)+2\rho )\right)+\beta \left(r^{-1}({\mathrm{diam}}_w\left(K\right)+2\rho )\right),$$

and $G^{\sim }$ is of bounded degree. □

Example 1.

Let $A=\left[\begin{array}{cc}4& 0\\ 0& 3\end{array}\right]$, and $\mathcal{D},{\mathcal{D}}^{\prime }$ be two digit sets as follows:

$$\begin{array}{ccc}& & \mathcal{D}=\{d_1,d_2,d_3,d_4\}=\left\{\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{c}1\\ 1\end{array}\right],\left[\begin{array}{c}2\\ 1\end{array}\right],\left[\begin{array}{c}3\\ 2\end{array}\right]\right\},\hfill \\ & & {\mathcal{D}}^{\prime }=\{d_1^{\prime },d_2^{\prime },d_3^{\prime },d_4^{\prime }\}=\left\{\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{c}3/4\\ 2/3\end{array}\right],\left[\begin{array}{c}2\\ 1\end{array}\right],\left[\begin{array}{c}3\\ 2\end{array}\right]\right\}.\hfill \end{array}$$

Define one IFS ${\left\{f_i\right\}}_{i=1}^4$ as $f_i\left(x\right)=A^{-1}(x+d_i)$ (see Figure 2a) and the other IFS ${\left\{g_i\right\}}_{i=1}^4$ as $g_i\left(x\right)=A^{-1}(x+d_i^{\prime })$ (see Figure 2b).

It is easy to verify that the IFS ${\left\{f_i\right\}}_{i=1}^4$ satisfies the OSC by taking the interior of the unit square as the open set, and the graph $G=({\Sigma }^{*},\mathcal{E})$ as of bounded degree. In fact, for any vertex $\mathbf{x}\in {\Sigma }^{*}$, the corresponding small rectangle $f_{\mathbf{x}}\left({[0,1]}^2\right)$ intersects at most 8 rectangles with the same size; hence, by the definition of the augmented tree, $\mathbf{x}$ has 1 father, 4 children, and at most 8 neighbors. Consequently, we have

$$deg\left(\mathbf{x}\right)\le 1+4+8=11.$$

According to Theorem 2, we know that the IFS ${\left\{g_i\right\}}_{i=1}^4$ satisfies the WSC, but not the OSC. Hence, the graph $G=({\Sigma }^{*},\mathcal{E})$ is not of bounded degree by Theorem 5. It can also be verified immediately. For $\mathbf{x}=14,\mathbf{y}=21$, we have $g_{\mathbf{x}}=g_{\mathbf{y}}$ (see Figure 2b). For any $k\ge 1$, let ${\mathcal{G}}_k=\{\mathbf{u}={\mathbf{x}}_1\cdots {\mathbf{x}}_k:{\mathbf{x}}_i=\mathbf{x}\mathit{or}\mathbf{y},i=1,\dots ,k\}.$ It follows that $g_{\mathbf{u}}=g_{\mathbf{v}}$ for any $\mathbf{u},\mathbf{v}\in {\mathcal{G}}_k$. Thus, the degree of the vertex $\mathbf{u}\in {\mathcal{G}}_k\subset {\Sigma }^{*}$ is at least $2^k-1$. Therefore, $G=({\Sigma }^{*},\mathcal{E})$ is not of bounded degree.

On the other hand, if we consider $\mathbf{x}=\{14,21\}$ as an equivalence class, i.e., a vertex in the quotient space ${\Sigma }^{*}/\sim$. Theorem 6 implies that $G^{\sim }:=({\Sigma }^{*}/\sim ,\mathcal{E})$ is of bounded degree.

Proof of Theorem 4. 

We first prove that the graph $G=({\Sigma }^{*},\mathcal{E})$ is a Gromov hyperbolic graph by contradiction. Suppose G is not hyperbolic, then by Lemma 1, for any integer $m\ge 1$, there exists a horizontal geodesic $\pi ({\mathbf{u}}_0,{\mathbf{u}}_1,\dots ,{\mathbf{u}}_{3m})$ with ${\mathbf{u}}_i\in {\Sigma }^n$ for some $n\ge 1$. The length of the geodesic is $3m$. We can find another path $p({\mathbf{u}}_0,{\mathbf{u}}_0^{-1},\dots ,o,\dots ,{\mathbf{u}}_{3m}^{-1},{\mathbf{u}}_{3m})$ from ${\mathbf{u}}_0$ to ${\mathbf{u}}_{3m}$, where o is the root of G. It follows that $3m\le 2n$, then $m<n$. We consider the set $\{{\mathbf{u}}_0^{-m},{\mathbf{u}}_1^{-m},\dots ,{\mathbf{u}}_{3m}^{-m}\}$, which is the m-th generation ancestor of $\pi ({\mathbf{u}}_0,{\mathbf{u}}_1,\dots ,{\mathbf{u}}_{3m})$. Definition 2 implies that either ${\mathbf{u}}_i^{-m}={\mathbf{u}}_{i+1}^{-m}$ or $({\mathbf{u}}_i^{-m},{\mathbf{u}}_{i+1}^{-m})\in {\mathcal{E}}_h$. Hence, there exists a shortest horizontal path, say, $p({\mathbf{v}}_0,\dots ,{\mathbf{v}}_l)$, joining ${\mathbf{u}}_0^{-m}$ and ${\mathbf{u}}_{3m}^{-m}$, where ${\mathbf{v}}_0={\mathbf{u}}_0^{-m},{\mathbf{v}}_l={\mathbf{u}}_{3m}^{-m}$, and ${\mathbf{v}}_j\in \{{\mathbf{u}}_0^{-m},{\mathbf{u}}_1^{-m},\dots ,{\mathbf{u}}_{3m}^{-m}\},i=0,1,\dots ,l$ (see Figure 3). Therefore, we obtain a new path from ${\mathbf{u}}_0$ to ${\mathbf{u}}_{3m}$ as follows:

$$\pi ({\mathbf{u}}_0,{\mathbf{u}}_0^{-1},\dots ,{\mathbf{u}}_0^{-m})\cup p({\mathbf{v}}_0,\dots ,{\mathbf{v}}_l)\cup \pi ({\mathbf{u}}_{3m}^{-m},\dots ,{\mathbf{u}}_{3m}^{-1},{\mathbf{u}}_{3m}).$$

By the property of geodesic, the length of path $p({\mathbf{v}}_0,\dots ,{\mathbf{v}}_l)$ is as follows:

$$l=\left|p\right({\mathbf{v}}_0,\dots ,{\mathbf{v}}_l\left)\right|\ge 3m-2m=m.$$

(7)

Through the following:

$$E=\bigcup _{i=0}^{3m}{K}_{{\mathbf{u}}_i},E^{\prime }=\bigcup _{i=0}^l{K}_{{\mathbf{v}}_i},$$

we can estimate the w-diameters of E and $E^{\prime }$. Since ${\mathbf{u}}_i\in {\Sigma }^n$ and $({\mathbf{u}}_i,{\mathbf{u}}_{i+1})\in {\mathcal{E}}_h$, we have ${\mathrm{diam}}_w\left(K_{{\mathbf{u}}_i}\right)\le r^n{\mathrm{diam}}_w\left(K\right)$ and $d_w(K_{{\mathbf{u}}_i},K_{{\mathbf{u}}_{i+1}})\le \rho r^n$. Hence, we determine the following:

$$\begin{array}{ccc}\hfill {\mathrm{diam}}_w\left(E\right)& \le & \sum _{i=0}^{3m}{\mathrm{diam}}_w\left(K_{{\mathbf{u}}_i}\right)+\sum _{i=0}^{3m-1}{d}_w(K_{{\mathbf{u}}_i},K_{{\mathbf{u}}_{i+1}})\hfill \\ & \le & (3m+1)r^n{\mathrm{diam}}_w\left(K\right)+3m\rho r^n\hfill \\ & <& (3m+1)({\mathrm{diam}}_w\left(K\right)+\rho )r^n.\hfill \end{array}$$

Since for any ${\mathbf{v}}_j$, there is i satisfying ${\mathbf{v}}_j={\mathbf{u}}_i^{-m}$, it follows that $K_{{\mathbf{u}}_i}\subset K_{{\mathbf{v}}_j}$, and $E\cap K_{{\mathbf{v}}_j}\ne \varnothing$. Then, considering the following:

$$\begin{array}{ccc}\hfill {\mathrm{diam}}_w\left(E^{\prime }\right)& \le & {\mathrm{diam}}_w\left(E\right)+2\underset{0\le i\le l}{max}\left\{{\mathrm{diam}}_w{K}_{{\mathbf{v}}_i}\right\}\hfill \\ & <& (3m+1)({\mathrm{diam}}_w\left(K\right)+\rho )r^n+2{\mathrm{diam}}_w\left(K\right)r^{n-m},\hfill \end{array}$$

we can choose m sufficiently large such that $(3m+1)({\mathrm{diam}}_w\left(K\right)+\rho )r^m<{\mathrm{diam}}_w\left(K\right)$. It is straightforward to obtain the following:

$${\mathrm{diam}}_w\left(E^{\prime }\right)<3{\mathrm{diam}}_w\left(K\right)r^{n-m}.$$

Let $E_0=A^{n-m}{E}^{\prime }$, then there exists a ball B with radius $3{\mathrm{diam}}_w\left(K\right)$ such that

$$E_0=\bigcup _{i=0}^l{A}^{n-m}{K}_{{\mathbf{v}}_i}\subset B.$$

Let $a_j\in A^{n-m}{K}_{{\mathbf{v}}_j},j=1,\dots ,l$ and let $l^{\prime }=\left[{\displaystyle \frac{l}{2}}\right]$, the largest integer that is not greater than ${\displaystyle \frac{l}{2}}$. By assumption, $p({\mathbf{v}}_0,\dots ,{\mathbf{v}}_l)$ is the shortest horizontal path from ${\mathbf{v}}_0$ to ${\mathbf{v}}_l$. Hence, $({\mathbf{v}}_i,{\mathbf{v}}_j)\notin {\mathcal{E}}_h$ for any $j>i+1$. It follows that $d_w(K_{{\mathbf{v}}_i},K_{{\mathbf{v}}_j})>\rho r^{n-m}$, as well as the following:

$$d_w(a_i,a_j)\ge d_w(A^{n-m}{K}_{{\mathbf{v}}_i},A^{n-m}{K}_{{\mathbf{v}}_j})>\rho .$$

By applying the pigeonhole principle, the ball B contains at least $l^{\prime }+1$ points $a_0,a_2,\dots ,a_{2l^{\prime }}$ such that the w-distance between any two of them is at least $\rho$. By (7), we have $l^{\prime }+1\ge {\displaystyle \frac{m}{2}}$, which is impossible, as m can be taken arbitrarily large. Hence, we have proven the hyperbolicity of G.

For any point $\eta \in \partial G$, we let $\pi [{\mathbf{u}}_0,{\mathbf{u}}_1,\dots ]$ be a geodesic ray representing $\eta$. Define a map $\psi :\partial G\to K$ as follows:

$$\psi \left(\eta \right)=\underset{n\to \infty }{lim}{f}_{{\mathbf{u}}_n}\left(x_0\right).$$

(8)

for some $x_0\in K$. The bijection of $\psi$ follows from [12] or [14] immediately. It remains to verify the Hölder inequality (4).

Let $\eta =\pi [{\mathbf{u}}_0,{\mathbf{u}}_1,\dots ],\xi =\pi [{\mathbf{v}}_0,{\mathbf{v}}_1,\dots ]$ be two different geodesic rays. There is a canonical bilateral geodesic $\zeta$ connecting $\eta$ and $\xi$, as follows:

$$\zeta =\pi [\dots ,{\mathbf{v}}_{n+1},{\mathbf{v}}_n,{\mathbf{t}}_1,\dots ,{\mathbf{t}}_l,{\mathbf{u}}_n,{\mathbf{u}}_{n+1},\dots ]$$

(9)

where ${\mathbf{v}}_n,{\mathbf{t}}_1,\dots ,{\mathbf{t}}_l,{\mathbf{u}}_n$ lie in the same level, say, ${\Sigma }^n$. Thus, we can deduce the following:

$$d_w(f_{{\mathbf{v}}_n}\left(x_0\right),f_{{\mathbf{u}}_n}\left(x_0\right))\le (l+2)r^n{\mathrm{diam}}_w\left(K\right)+(l+1)\rho r^n.$$

Note that for any $m\ge 0$, we have :

$$\begin{array}{c}\hfill \psi \left(\xi \right)\in K_{{\mathbf{v}}_{n+m}}\subset K_{{\mathbf{v}}_n}\mathrm{and}\psi \left(\eta \right)\in K_{{\mathbf{u}}_{n+m}}\subset K_{{\mathbf{u}}_n}.\end{array}$$

(10)

which implies that

$$d_w(\psi \left(\eta \right),f_{{\mathbf{u}}_n}\left(x_0\right))\le r^n{\mathrm{diam}}_w\left(K\right),d_w(\psi \left(\xi \right),f_{{\mathbf{v}}_n}\left(x_0\right))\le r^n{\mathrm{diam}}_w\left(K\right).$$

By making use of Proposition 1 (iii) twice, we can conclude the following:

$$\begin{array}{ccc}\hfill d_w(\psi \left(\eta \right),\psi \left(\xi \right))& =& w(\psi \left(\eta \right)-f_{{\mathbf{u}}_n}\left(x_0\right)+f_{{\mathbf{u}}_n}\left(x_0\right)-f_{{\mathbf{v}}_n}\left(x_0\right)+f_{{\mathbf{v}}_n}\left(x_0\right)-\psi \left(\xi \right))\hfill \\ & \le & C_0{r}^n,\hfill \end{array}$$

where $C_0>0$ is a constant.

By (5) and (9), the Gromov product $|\eta \wedge \xi |=n-{\displaystyle \frac{l+1}{2}}$. Thus, we have the following:

$${\theta }_{\tau }(\eta ,\xi )=exp(-\tau |\eta \wedge \xi \left|\right)=exp\{-\tau (n-{\displaystyle \frac{1}{2}}(l+1))\}.$$

By Lemma 1, the length $l+1$ of the horizontal geodesic $\pi [{\mathbf{v}}_n,{\mathbf{t}}_1,\dots ,{\mathbf{t}}_l,{\mathbf{u}}_n]$ is bounded by $\kappa$. It implies that there is $C_1>0$, resulting in the following:

$$C_{1}exp(-\tau n)\le {\theta }_{\tau }(\eta ,\xi )\le C_{1}exp(-\tau n).$$

Hence, we determine the following:

$$d_w(\psi \left(\eta \right),\psi \left(\xi \right))\le C_2{\theta }_{\tau }^{\alpha }(\eta ,\xi )$$

where $C_2>0,\alpha ={\displaystyle \frac{logq}{d\tau }}$.

On the other hand, it is clear that $({\mathbf{v}}_{n+1},{\mathbf{u}}_{n+1})\notin {\mathcal{E}}_h$. Then, $d_w(K_{{\mathbf{v}}_{n+1}},K_{{\mathbf{u}}_{n+1}})>\rho r^{n+1}$. Using (10) again, there is $C_3>0$, allowing us to achieve the following:

$$d_w(\psi \left(\eta \right),\psi \left(\xi \right))\ge d_w(K_{{\mathbf{v}}_{n+1}},K_{{\mathbf{u}}_{n+1}})\ge \rho rexp(-\tau n\alpha )\ge C_3{\theta }_{\tau }^{\alpha }(\eta ,\xi ).$$

Therefore, we obtain (4) and complete the proof. □

4. Conclusions

In this paper, we demonstrate the close relationship between a self-affine IFS and an augmented tree induced by the IFS. First, we propose a novel characterization of the OSC and WSC in the IFS by using the bounded degree of the augmented tree, providing a unified framework to analyze separation properties of IFSs via graph-theoretic invariants. Second, we show that the augmented tree is a Gromov hyperbolic graph, and its hyperbolic boundary is Hölder equivalent to the self-affine set generated by the IFS, establishing a canonical association between self-affine IFSs and Gromov hyperbolic graphs.