Generalized Dimensions of Self-Affine Sets with Overlaps

Let $M_d(\mathbb{R})$ be the set of all $d\times d$ matrices with entries in $\mathbb{R}$, and let $A\in M_d(\mathbb{R})$ be an expanding matrix. Let $N\ge 2$ and $n_1,\dots ,n_N$ be positive integers. Write $A_j=A^{n_j},1\le j\le N$ and $\mathcal{D}=\{d_1,\dots ,d_N\}\subset {\mathbb{R}}^d$. Then, we can define a self-affine iterated function system (IFS) ${\{S_j\}}_{j=1}^N$ on ${\mathbb{R}}^d$ by

According to [1], there exists a unique nonempty compact set $E:=E({\{A_j\}}_{j=1}^N,\mathcal{D})$ such that

We call E a self-affine set. Without loss of generality, we always assume that $d_1=0$ and $n_1\ge n_2\ge \cdots \ge n_N$ throughout the paper.

The dimensional theory of self-affine IFS is a major topic in fractal geometry and dynamical systems. Many important results have been achieved in this area. In the case of self-similar IFS, i.e., $A_j$ in (1) are similitudes as $A_j^{-1}={\rho }_j{R}_j$, where $0<{\rho }_j<1$ and $R_j$ are orthonormal matrices, the set E (2) is usually called a self-similar set. If the open set condition holds, the Hausdorff and box dimensions of E are clear, which satisfy ${dim}_{H}E={dim}_{B}E=\alpha$, where $\alpha$ is the unique solution to the equation ${\sum }_{j=1}^N{\rho }_j^s=1$ (see refs. [1,2,3]). The IFS (1) is said to satisfy the open set condition (OSC) [1,3] if there exists a bounded nonempty open set $U\subset {\mathbb{R}}^d$ such that $U\supset {\bigcup }_{j=1}^N{S}_j(U)$ with disjoint union.

If the OSC does not hold, overlaps may occur; it is very challenging to obtain a simple dimensional formula. In 2001, Ngai and Wang [4] defined a finite type condition to deal with the self-similar IFSs with overlaps and described an algorithm for the dimension of self-similar sets. Subsequently, several other types of separation conditions were developed as well, e.g., weak separation condition, generalized finite type condition. With these conditions, it becomes easy to find the dimension of self-similar sets with overlapping structure (see refs. [5,6,7,8,9]). On the other hand, from the viewpoint of dynamical systems, the dimensional results of invariant measures or self-similar sets are closely related to a Ledrappier–Young type formula (see, e.g., [10] and references therein).

However, compared with self-similar sets, research on fractal dimension of self-affine sets has been progressing very slowly (see a survey paper [11]). Except for some special cases, such as the Bedford–McMullen set [12,13], it is difficult to give an exact dimensional formula (see refs. [14,15]).

Recently, by employing a pseudo-norm w rather than the Euclidean norm, He and Lau [16] introduced the generalized Hausdorff measure ${\mathcal{H}}_w^t$, the generalized Hausdorff dimension ${dim}_H^w$ and the generalized box dimension ${dim}_B^w$ (see definitions in Section 2). The w is determined by the given matrix A. The generalized dimensions are useful to estimate the exact fractal dimension of self-affine sets. Under the OSC, He and Lau obtained a formula of the generalized dimensions on a class of self-affine IFSs with equal linear parts (i.e., all $A_j\equiv A$). Later, based on He and Lau’s results, Fu, Gabardo and Qiu [17] further proved that the self-affine IFS satisfies the OSC if and only if the generalized Hausdorff measure ${\mathcal{H}}_w^t(E)>0$. The second author and Yang [18] also computed the generalized dimensions of the attractors of a class of self-affine graph-directed IFSs.

Motivated by the above studies, in this paper, we try to develop Ngai and Wang’s idea and define a new finite type condition on the self-affine IFS (1). Then, we generalize He and Lau’s dimensional result onto the overlapping situation.

Let $\Sigma =\{1,2,\dots ,N\},{\Sigma }^k=\{j_1{j}_2\dots j_k:j_i\in \Sigma ,1\le i\le k\},{\Sigma }^{\ast }={\bigcup }_{k=0}^{\infty }{\Sigma }^k$ with ${\Sigma }^0=\{\varnothing \}$ and ${\Sigma }^{\infty }=\{j_1{j}_2\cdots :j_i\in \Sigma ,i\ge 1\}$. If $I=i_1\dots i_h$ and $J=j_1\dots j_{\ell }$; we denote by $IJ=i_1\dots i_h{j}_1\dots j_{\ell }$ the concatenation of them. If $I=i_1{i}_2\cdots \in {\Sigma }^{\infty }$, let ${I|}_k=i_1\dots i_k$ be the restriction of the first k symbols of I. For any $J=j_1\dots j_k\in {\Sigma }^k$, we denote $|J|=k$ the length of J, and write $S_J(x)=S_{j_1}\circ \cdots \circ S_{j_k}(x)$, $A_J={\prod }_{i=1}^k{A}_{j_i}=A^{j_1+\cdots +j_k}.$

Assume $|detA|=q$ and write $r_j=q^{-n_j/d},r_J=r_{j_1}\dots r_{j_k}$ which stand for the contraction ratios of the maps $S_j$ and $S_J$ under w, respectively (see Section 2). For any $k\ge 0$, we denote

Let ${\mathcal{V}}_0=\{(\mathrm{id},0)\},{\mathcal{V}}_k=\{(S_J,k):J\in {\Lambda }_k\}\mathrm{for}k\ge 1\mathrm{and}\mathcal{V}={\bigcup }_{k\ge 0}{\mathcal{V}}_k$. Let U be an invariant open set of the IFS (1), i.e., U satisfies ${\bigcup }_{j=1}^N{S}_j(U)\subset U$. We say that $\mathbf{u}=(S_I,k),\mathbf{v}=(S_J,k)\in {\mathcal{V}}_k$ are neighbors with respect to U if $S_I(U)\cap S_J(U)\ne \varnothing$. The neighborhood of $\mathbf{v}$ is defined as the set of all its neighbors. Two neighborhoods are said to be of the same type if they are the same in the sense of scaling and translation.

We say an IFS (1) satisfies the finite type condition (FTC) when its neighborhood types are finite. Every FTC determines an incidence matrix (see the details in Section 3). Now we turn to present the key conclusion on dimensions.

From the definition, it seems that the FTC depends heavily on the invariant open set. However, the choice of open sets can be more flexible. Let V be a bounded invariant set of ${\{S_j\}}_{j=1}^N$ and define

This paper is organized as follows. In Section 2, we give known results on generalized dimensions. In Section 3, we define the FTC of self-affine IFSs and prove Theorem 1. In Section 4, we provide some sufficient conditions for the FTC to hold, and prove Theorem 2. In Section 5, we include an illustrative example on the computation of generalized dimensions of self-affine sets.

Following the notation in [16], let $A\in M_d(\mathbb{R})$ be an expanding matrix with $|detA|=q$, and $B(x,r)$ the closed ball with center x and radius r. Set the region $V=A(B(0,1))\backslash B(0,1)$. Take any $0<\delta <{\displaystyle \frac{1}{2}}$ and any positive smooth even function ${\varphi }_{\delta }$ supported on $B(0,\delta )$ with $\int {\varphi }_{\delta }(x)dx=1$. We define a function w by where ${\chi }_V$ is the characteristic function of V and ${\chi }_V\ast {\varphi }_{\delta }(·)$ means the convolution. The $w(·)$ is called a pseudo-norm on ${\mathbb{R}}^d$.

For $F\subset {\mathbb{R}}^d$, ${\mathrm{diam}}_{w}F=sup\{w(x-y):x,y\in F\}$ stands for the w-diameter of F and $B_w(x,r)=\{y:w(x-y)\le r\}$ a w-ball.

It is worth mentioning that property (iii) implies that the matrix A is a similitude under w. This fact plays an important role in the present paper.

For $\alpha >0$, the $\alpha$-dimensional generalized Hausdorff measure of F with respect to w is given by where

Accordingly, we define the generalized box dimension. Denote by ${\overline{dim}}_B^{w}F,{\underline{dim}}_B^{w}F$ the upper and lower box dimensions of F under w. If the two values coincide, we say that the generalized box dimension of F exists, denoted by ${dim}_B^{w}F$. The following consequence is trivial.

A simple relationship between the generalized dimensions is as follows.

We first introduce the FTC of the IFS (1), then consider the generalized dimension of the self-affine set (2).

Recall that $q=|detA|$, $r_j=q^{-n_j/d}$ for $j=1,\dots ,N$, and $r_1\le \cdots \le r_N$ (as $n_1\ge n_2\ge \cdots \ge n_N$). Let

Define

For $\mathbf{v}=(S_J,k)\in {\mathcal{V}}_k$, we write $S_{\mathbf{v}}=S_J$ and $r_{\mathbf{v}}=r_J$, and define a map $\pi :\Lambda \to \mathcal{V}$ by

Let $U\subset {\mathbb{R}}^d$ be a bounded invariant set under ${\{S_j\}}_{j=1}^N$, i.e., ${\bigcup }_{j=1}^N{S}_j(U)\subset U$. Define

The set $N_U(\mathbf{v})$ is named as the neighborhood of $\mathbf{v}$ involving U. $\mathbf{v}\in {\mathcal{V}}_k$ and $\mathbf{u}\in {\mathcal{V}}_{\ell }$ are called equivalent, denoted by $\mathbf{v}\sim \mathbf{u}$, if $\phi =S_{\mathbf{v}}\circ S_{\mathbf{u}}^{-1}$ is of the form $\phi (x)=A^{-(k-\ell )n_1}x+c$ such that

We use $[\mathbf{v}]$ to represent the equivalence class of $\mathbf{v}$ under the relation ∼. Trivially, $\mathbf{v}\sim \mathbf{u}$ if and only if

The major restriction of Ngai and Wang’s FTC in [4] is that the contraction ratios of similitudes must be commensurable. In the present setting, the affine maps $S_j$’s act as similitudes under the pseudo-norm w. More precisely, note that the linear part of $S_j$ is $A_j(=A^{n_j})$. Hence, the contraction ratio of each $S_j$ is $r_j:=q^{-n_j/d}$ under the pseudo-norm w. Clearly, $r_j,j=1,\dots ,N$, are exponentially commensurable. In this sense, the FTC is well defined.

Suppose that the IFS ${\{S_j\}}_{j=1}^N$ satisfies the FTC for open set U. Geometrically, iterates of U under $S_I(I\in {\Lambda }_k)$ generate a neighborhood system ${\mathcal{N}}_U(\mathbf{v})$ (where $\mathbf{v}=(S_I,k)$). FTC states that there are only finitely many distinct classes of neighborhood systems. Hence, it allows us to set up a directed graph which yields an incidence matrix T to count the number of distinct iterates. The following is a standard process to construct the directed graph and the incidence matrix.

Algorithm for constructing the directed graph:

Step I: For a vertex $\mathbf{v}=(S_I,k)\in {\mathcal{V}}_k$, if there exists a vertex $\mathbf{u}$ in ${\mathcal{V}}_{k+1}$ of the form $(S_I{S}_J,k+1)$ for some $J\in {\Sigma }^{\ast }$, we call $\mathbf{u}$ an offspring generated by $\mathbf{v}$. From $\mathbf{v}$ to $\mathbf{u}$, we label an edge: $\mathbf{v}\stackrel{J}{\to }\mathbf{u}$, where J is the label of the edge.

Step II: Due to the overlaps, it is possible to have more than one $\mathbf{v}\in {\mathcal{V}}_k$ generating a common offspring. So we need the lexicographical order for ${\Sigma }^{\ast }$ to obtain a reduced graph. For each vertex $\mathbf{u}\in {\mathcal{V}}_{k+1}$, let ${\mathbf{v}}_1,\dots ,{\mathbf{v}}_p$ be all the vertices in ${\mathcal{V}}_k$ that generate the offspring $\mathbf{u}$, with ${\mathbf{v}}_{\ell }\stackrel{J_{\ell }}{\to }\mathbf{u},1\le \ell \le p$. If $J_1<\cdots <J_p$ in the lexicographical order, we hold the smallest edge $J_1$ and eliminate all the other edges. Denote by $\Gamma$ the edge set of the resulting graph. Therefore, we obtain a reduced graph $(\mathcal{V},\Gamma )$ such that each vertex of $\mathcal{V}$ has a unique parent.

In the reduced graph $(\mathcal{V},\Gamma )$, $\mathbf{v}{⟶}_{\mathcal{V}}\mathbf{u}$ means that $\mathbf{u}$ is an offspring of $\mathbf{v}$. A path in $(\mathcal{V},\Gamma )$ is a sequence $\left({\mathbf{v}}_0,{\mathbf{v}}_1,{\mathbf{v}}_2,\dots \right)$ such that ${\mathbf{v}}_j\in {\mathcal{V}}_j$ and ${\mathbf{v}}_j{⟶}_{\mathcal{V}}{\mathbf{v}}_{j+1}$ for all $j⩾0$, where ${\mathbf{v}}_0=(\mathrm{id},0)$ is the root. Let $\mathcal{B}$ be the set of all paths in $(\mathcal{V},\Gamma )$. For the given vertices ${\mathbf{v}}_0,{\mathbf{v}}_1,\dots ,{\mathbf{v}}_k$ such that ${\mathbf{v}}_j{⟶}_{\mathcal{V}}{\mathbf{v}}_{j+1}$, we define a branch as follows:

Finally, we evaluate $\#{\mathcal{V}}_k$ by the incidence matrix T. Note that ${\mathcal{V}}_0=\{(\mathrm{id},0)\}$ and $\mathcal{V}/\sim =\{[{\mathbf{v}}_1],\dots ,[{\mathbf{v}}_m]\}$. We may assume that $(\mathrm{id},0)\in [{\mathbf{v}}_1]$. According to the reduced graph $(\mathcal{V},\Gamma )$, we have where $\theta ={(1,1,\dots ,1)}^t,e_1={(1,0,\dots ,0)}^t\in {\mathbb{R}}^m$.

It seems that the definition of FTC in the previous section depends heavily on the choice of open sets. Actually, the choice can be quite flexible. In this section, we discuss this problem.

We modify the notion of FTC to a slightly different form which is more convenient to use. Let V be a bounded invariant set of ${\{S_j\}}_{j=1}^N$, and $\mathcal{F}(V)$ as in (3). We can define the relation ${\sim }_V$ as in (6) by replacing U with V. Then, the following result is straightforward.

We provide an example in this last section to illustrate our main results.

Fix $U=(0,1)\times (0,1)$ as an invariant open set, and let ${\mathcal{T}}_1$ be the equivalence type of ${\mathbf{v}}_0=(\mathrm{id},0)$. The iterates of U under $S_1,S_2,S_3$ are as shown in Figure 1. It is easy to check that and where $S_1={(x/9,y/4)}^t, S_2={(x/9+8/9,y/4+3/4)}^t, S_{31}={(x/27+2/27,y/8+1/8)}^t,$$S_{32}={(x/27+10/27,y/8+1/2)}^t, S_{33}={(x/9+8/81,y/4+3/16)}^t.$