Distribution of $\delta$-connected components of self-affine sponge of Lalley-Gatzouras type

TL;DR

This paper extends the understanding of the distribution of $oldsymbol{ ext{delta}}$-connected components in self-affine Lalley-Gatzouras sponges, establishing a measure-theoretic relation similar to known results for other fractals.

Contribution

It generalizes the relation between $oldsymbol{h_E(oldsymbol{ ext{delta}})}$ and $oldsymbol{ ext{delta}^{- ext{dim}_B E}}$ to Lalley-Gatzouras type self-affine sponges, introducing a Bernoulli measure framework.

Findings

Existence of a Bernoulli measure $oldsymbol{ extmu}$ for Lalley-Gatzouras sponges.

Asymptotic relation $oldsymbol{h_R(oldsymbol{ ext{delta}}) oldsymbol{ extasymp} oldsymbol{ extmu}(R) oldsymbol{ ext{delta}^{- ext{dim}_B E}}$ as $oldsymbol{ ext{delta} o 0}$.

Generalization of known fractal component distribution results to a broader class of self-affine sets.

Abstract

Let $(E, \rho)$ be a metric space and let $h_E\left( \delta \right)$ be the cardinality of the set of $\delta$-connected components of $E$. In literature, in case of that $E$ is a self-conformal set satisfying the open set condition or $E$ is a self-affine Sierpi\'nski sponge, necessary and sufficient condition is given for the validity of the relation $ h_E(\delta)\asymp \delta^{-\dim_B E}, \text{ when }\delta\to 0. $ In this paper, we generalize the above result to self-affine sponges of Lalley-Gatzouras type; actually in this case, we show that there exists a Bernoulli measure $\mu$ such that for any cylinder $R$, it holds that $ h_R(\delta)\asymp \mu(R) \delta^{-\dim_B E}, \text{ when }\delta\to 0. $

Equations

$$h_E(\delta )\asymp {\delta }^{-{\dim }_{B}E},\text{ when }\delta \to 0.$$

$$h_R(\delta )\asymp \mu (R){\delta }^{-{\dim }_{B}E},\text{ when }\delta \to 0.$$

$$g_E(n)\asymp n^{-1/{\dim }_{B}E},n\ge 1,$$

$$g_E(n)\asymp n^{-1/\gamma }\iff h_E(\delta )\asymp {\delta }^{-\gamma }.$$

$$2\le n_1\le n_2<\cdots \le n_d$$

$${\pi }_j(x_1,\dots ,x_d)=(x_1,\dots ,x_j).$$

$$N_A(\delta )$$

$${\dim }_{B}A=\underset{\delta \to 0}{\lim }-\frac{\log N_A(\delta )}{\log \delta }$$

$$M_0^{-1}\mu (R){\delta }^{-{\dim }_B\Lambda }\le N_R(\delta )\le M_0\mu (R){\delta }^{-{\dim }_B\Lambda }.$$

$$M^{-1}\mu \left(R\right){\delta }^{-{\dim }_B\Lambda }\le h_R\left(\delta \right)\le M\mu \left(R\right){\delta }^{-{\dim }_B\Lambda },$$

$$h_R\left(\delta \right)=O\left(\mu \left(R\right){\delta }^{-\chi }\right),\delta \to 0.$$

$$h_R(\delta )\asymp N_R(\delta ),$$

$$\left||S^{\prime }(x)|-|S^{\prime }(y)|\right|\le C|x-y{|}^{\alpha }\text{ for all }x,y\in V.$$

$$C_1^{-1}|S_{\mathbf{\sigma }}(E)|\cdot |x-y|\le |S_{\mathbf{\sigma }}(x)-S_{\mathbf{\sigma }}(y)|\le C_1|S_{\mathbf{\sigma }}(E)|\cdot |x-y|$$

$$C_2^{-1}|K_{\mathbf{\sigma }}{|}^s\le \mu (K_{\mathbf{\sigma }})\le C_2|K_{\mathbf{\sigma }}{|}^s.$$

$$\begin{array}[]{rlr}h_{\delta}(K_{I})&\geq h_{\delta^{\prime}}(K)&\text{(By the similarity mentioned above.)}\\ &\geq M(\delta^{\prime})^{-s}&\text{(By the maximal power law.)}\\ &=MC_{1}^{-s}|K_{I}|^{s}\delta^{-s}&\\ &\geq M(C_{1}C_{2})^{-s}\mu(K_{I})\delta^{-s}.&\text{(By Lemma \ref{lem:Alfors}.)}\end{array}$$

$${\varphi }_a\left(x_1,\cdots x_d\right)=\left({\varphi }_{a,1}\left(x_1\right),\cdots ,{\varphi }_{a,d}\left(x_d\right)\right)$$

$${\varphi }_a\left({\left[0,1\right]}^d\right)=\prod _{i=1}^d{\varphi }_{a,i}\left(\left[0,1\right]\right)\in {\left[0,1\right]}^d$$

$$\left[0,1\right]={\varphi }_{0,i}\left[0,1\right)\cup \cdots \cup {\varphi }_{n_i-1,i}\left[0,1\right)$$

$${\varphi }_{a,1}^{\prime }>\cdots >{\varphi }_{a,d}^{\prime },\text{ for all }a\in \mathcal{D},$$

$${\Phi }_{\left\{1,\cdots ,j\right\}}={\left(\left({\varphi }_{a,1},\cdots ,{\varphi }_{a,j}\right)\right)}_{a\in {\pi }_j\left(\mathcal{D}\right)},$$

$${\left\{{\varphi }_{\mathbf{d},\{1,\dots ,j\}}({\mathbb{I}}^j)\right\}}_{\mathbf{d}\in {\pi }_j(\mathcal{D})}$$

$${\mu }_{\mathbf{p}}(\cdot )=\sum _{\mathbf{d}\in \mathcal{D}}{p}_{\mathbf{d}}{\varphi }_{\mathbf{d}}^{-1}(\cdot )$$

$$\sum _{f_1\in {\Phi }_{\{1\}}}(f_1^{\prime }{)}^{{\beta }_1}=1.$$

$$\sum _{(f_1,\dots ,f_j)\in {\Phi }_{\{1,\dots ,j\}}}\prod _{k=1}^j(f_k^{\prime }{)}^{{\beta }_k}=1.$$

$$p_{\mathbf{f}}=\prod _{k=1}^j(f_k^{\prime }{)}^{{\beta }_k}.$$

$${\dim }_B{\Lambda }_{\Phi }=\sum _{j=1}^d{\beta }_j,$$

$${\alpha }_j:={\dim }_B{\Lambda }_j=\sum _{k=1}^j{\beta }_k.$$

$$S(W)=\prod _{j=1}^k{\varphi }_{i_j,d}^{\prime }$$

$$r_{\ast }=\min \left\{{\varphi }_{a,d}^{\prime };a\in \mathcal{D}\right\}.$$

Full text

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In this paper, we generalize the above result to self-affine sponges of Lalley-Gatzouras type; actually in this case, we show that there exists a Bernoulli measure $\mu$ such that for any cylinder $R$, it holds that

[TABLE]

The work is supported by the start-up research fund from Huzhou University No. RK21089.

2010 Mathematics Subject Classification: 28A80, 28A78.

Key words and phrases:  self-affine sponge, maximal power law, component-counting measure.

• The correspondence author.

1. Introduction

Let $\left(E,\rho\right)$ be a metric space and let $\delta>0$. Two points $x,y\in E$ are said to be $\delta$-equivalent if there exists a sequence $\left\{x_{1}=x,x_{2},...,x_{k-1},x_{k}=y\right\}\subset E$ such that $\rho\left(x_{i},x_{i+1}\right)\leq\delta$ for $1\leq i\leq k-1$. A $\delta$-equivalent class of $E$ is called a $\delta$-connected component of $E$. We denote by $h_{E}\left(\delta\right)$ the cardinality of the set of $\delta$-connected components of $E$.

A notion closely related to $\delta$-connected component is the gap sequence of a compact metric space, which has been studied by many mathematicians; see for instance, [5, 9, 17, 30, 7]. Some early works ([17, 9, 7]) observed that for some totally disconnected self-similar sets and self-conformal sets $E$, the gap sequence, which we denote by $\{g_{E}(n)\}_{n\geq 1}$, is comparable to $n^{-1/\dim_{B}E}$, which is written as

[TABLE]

where $\dim_{B}$ denotes the box dimension. (Two functions $f,g:X\to{\mathbb{R}}$ are said to be comparable, denoted by $f\asymp g$, if there exists a constant $c>0$ such that $c^{-1}g(x)\leq f(x)\leq cg(x)$ for all $x\in X$.)

Let $(E,\rho)$ be a compact metric space and let $\gamma>0$. It is shown (Miao, Xi and Xiong [25], Zhang and Huang [34]) that

[TABLE]

Motivated by this relation, Zhang and Huang [34] proposed the following definition.

Definition 1.1**.**

Let $E$ be a compact metric space. We say $E$ satisfies the power law with index $\gamma$ if $h_{E}\left(\delta\right)\asymp\delta^{-\gamma}$; if $\gamma=\dim_{B}E$, in addition, we say $E$ satisfies the maximal power law.**

There are many works devoted to the maximal power law of attractors of IFS ([17, 9, 7, 25, 20, 21, 13, 34]). An iterated function system (IFS) is a family of contractions $\Phi=\{\varphi_{j}\}_{j=1}^{N}$ on a metric space $X$. In this paper, we will always assume that all $\varphi_{j}$ are injections. The attractor of the IFS is the unique nonempty compact set $E$ satisfying $E=\bigcup_{j=1}^{N}\varphi_{j}(E)$; especially, it is called a self-similar set if all $\varphi_{j}$ are similitudes. An IFS $\{\varphi_{j}\}_{j=1}^{N}$ is said to satisfy the open set condition (OSC), if there is a bounded nonempty open set $U\subset\mathbb{R}^{d}$ such that for all $1\leq i\leq N$, $\varphi_{i}(U)\subset U$ and $\varphi_{i}(U)\cap\varphi_{j}(U)=\emptyset$ for $1\leq i\neq j\leq N$; moreover, if $U\cap E\neq\emptyset$, then we say $E$ satisfies the strong open set condition (SOSC) and we call $U$ a feasible strong open set for $E$.

Lapidus, Pomerance [17] and Falconer [9] confirmed maximal power law for self-similar sets $E\subset\mathbb{R}$ satisfying the strong separation condition, and Deng, Wang and Xi [7] generalized this result to the self-conformal sets. Recently, Huang and Zhang [13] gave a complete answer in case of the self-conformal set satisfying the open set condition.

A point $x\in E$ is called a trivial point of $E$ if $\{x\}$ is a connected component of $E$.

Proposition 1.1** ([13]).**

Let $K$ be a $\mathcal{C}^{1+\alpha}$ self-conformal set generated by the IFS $\{S_{j}\}_{j=1}^{N}$. Assume $K$ satisfies the OSC. Then the following statements are equivalent:

(i)* $K$ satisfies the maximal power law.*

(ii)* There exists a strong open set $U$ such that $U$ contains trivial points of $K$.*

(iii)* Every strong open set for $K$ contains trivial points of $K$.*

(iv)* The set of trivial points of $K$ is dense in $K$.*

There are several works devoted to the maximal power law of Bedford-McMullen carpets[25, 20, 21] and self-affine Sierpiński sponge [34]. Let $d\geq 2$ and let

[TABLE]

be a sequence of integers. Let $\Gamma=\text{diag}(n_{1},\dots,n_{d})$ be the $d\times d$ diagonal matrix. Let ${\mathcal{D}}=\{{\mathbf{i}}_{1},\dots,{\mathbf{i}}_{N}\}\subset\prod_{j=1}^{d}\{0,1,\dots,n_{j}-1\}$. For ${\mathbf{i}}\in{\mathcal{D}}$ and $z\in\mathbb{R}^{d}$, we define $S_{{\mathbf{i}}}(z)=\Gamma^{-1}(z+{\mathbf{i}})$. The attractor of the IFS $\{S_{{\mathbf{i}}}\}_{{\mathbf{i}}\in{\mathcal{D}}}$ , which is denoted by $E=K(\Gamma,{\mathcal{D}})$, is called a $d$-dimensional generalized self-affine Sierpi$\acute{\text{n}}$ski sponge (see Olsen [26]). The set $E$ is called a fractal cube if $n_{1}=n_{2}=\cdots=n_{d}$; it is called a self-affine Sierpi$\acute{\text{n}}$ski sponge if $n_{1}<n_{2}<\cdots<n_{d}$ (see Kenyon and Peres [14]), and especially when $d=2$, $E$ is called a Bedford-McMullen carpet.

We say $E$ is non-degenerated if $E$ is not contained in a $(d-1)$-dimensional face of the cube $[0,1]^{d}$. For $1\leq j\leq d$, let $\pi_{j}:\mathbb{R}^{d}\to\mathbb{R}^{j}$ be the projection

[TABLE]

Proposition 1.2** ([34]).**

Let $E$ be a non-degenerated self-affine Sierpi$\acute{\text{n}}$ski sponge. Then $E$ satisfies the maximal power law if and only if $E$ and all $\pi_{j}(E)(1\leq j\leq d-1)$ possess trivial points.

The main purpose of the present paper is to prove a stronger version of Proposition 1.2 for Lalley-Gatzouras sponges. A diagonal self-affine sponge $\Lambda$ is said to be a Lalley-Gatzouras sponge if the generating IFS satisfies a coordinate ordering condition as well as a neat projection condition ([16, 6]). See Section 3 for the precise definitions.

Remark 1.1**.**

The diagonal self-affine sponge is an important class of self-affine sets received a lot of studies in recent years, Baranski [1], Das and Simmons [6], Feng and Wang [10], Mackay [23], Peres [28] Banaji and Kolossvary [2] on dimension theory, King [15], Jordan and Rams [22], Barrel and Mensi[3], Olsen [26], Reeve[32] on multifractal analysis, Li et al [19], Rao et al [31], Yang and Zhang [33], on Lipschitz classification, etc. The distribution of $\delta$-connected components illustrates the metric and topology properties of the self-affine sponges from a new point of view.

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However, this generalization is not trivial. The difficulty comes from the fact that given a cylinder $E_{I}$ of a Lalley-Gatzouras type sponge $E$, the relation between $h_{E}(\delta)$ and $h_{E_{I}}(\delta)$ is unclear. In this paper, we overcome this difficulty by using a recent result of Huang et al [12] on box-counting measures.

Denote $\Sigma=\{1,\dots,N\}$ and set $\Sigma^{*}=\bigcup_{n\geq 0}\Sigma^{n}$. For $I=i_{1}\dots i_{n}\in\Sigma^{*}$, we call $E_{I}=\varphi_{I}(E)=\varphi_{i_{1}}\circ\dots\circ\varphi_{i_{n}}(E)$ an $n$-th cylinder of $E$. We call $\delta(z+[0,1]^{d})$ a $\delta$-mesh box if $z\in\mathbb{Z}^{d}$. For $A\subset\mathbb{R}^{d}$, we use

[TABLE]

to denote the number of $\delta$-mesh boxes intersecting $A$. It is well known that

[TABLE]

if the limit exists (see for instance, [8]).

For a Lalley-Gatzouras sponge, a special Bernoulli measure, which we will call the canonical Bernoulli measure (see Section 3 for precise definition), is closely related to the box-dimension ( [16, 12]). Recently, Huang et al proved the following result.

Proposition 1.3**.**

([12]) Let $\Lambda$ be a Lalley-Gatzouras sponge. Then the canonical Bernoulli measure $\mu$ is a cylinder box-counting measure in the sense that, there is a constant $M_{0}>0$ such that for any cylinder $R$ of $\Lambda$, and any $\delta<S(R)$, the shortest side of $R$, it holds that

[TABLE]

Motivated by the above result, we propose the following concept.

Definition 1.2** (Component-counting measure).**

Let $\Lambda$ be the attractor of an IFS $\Phi$, and let $\mu$ be a finite Borel measure on $\Lambda$. If there is a constant $M\geq 1$ such that for any cylinder $R$ of $\Lambda$, there exists a $\delta_{0}=\delta_{0}(R)$ such that

[TABLE]

*for $\delta<\delta_{0}$, then we call $\mu$ a component-counting measure of $\Lambda$ .

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Remark 1.2**.**

Clearly, if $\Lambda$ admits a component-counting measure, then not only $\Lambda$, but all its cylinders satisfies the maximal power law with a uniform constant $M$. This reflects a kind of homogenity of $\Lambda$.**

First, we add one more equivalent statement to the list in Proposition 1.1.

Theorem 1.1**.**

Let $K$ be a ${\mathcal{C}}^{1+\alpha}$ self-conformal set satisfying the OSC, and let $\mu$ be the Gibbs measure. Then $K$ satisfies the maximal power law if and only if $\mu$ is a component-counting measure.

A Lalley-Gatzouras sponge $\Lambda$ is said to be degenerated if $\Lambda$ is contained in a face of $\left[0,1\right]^{d}$ with dimension $d-1$. From now on, we will always assume that $\Lambda$ is a non-dengerated Lalley-Gatzouras sponge. Under this assumption, $\Lambda$ satisfies the strong open set condition with the open set $V=(0,1)^{d}$. Denote $\Lambda_{j}=\pi_{j}\left(\Lambda\right)$ where $\pi_{j}$ is defined by (1.1); clearly $\Lambda_{j}$ is also non-degenerated, and $V_{j}=(0,1)^{j}$ is a feasible strong open set for $\Lambda_{j}$. (See Lemma 4.4.) A trivial point $x$ of $\Lambda$ is called an inner trivial point of $\Lambda$ if $x\in(0,1)^{d}$.

The main result of the present paper is the following two theorems.

Theorem 1.2**.**

Let $\Lambda$ be a non-degenerated Lalley-Gatzouras type sponge, and let $\mu$ be the canonical Bernoulli measure. Then the following statements are equivalent

(i) $\mu$ is a component-counting measure.

(ii) $\Lambda$ satisfies the maximal power law.

(iii) $V_{j}=(0,1)^{j}$ possesses trivial points of $\Lambda_{j}=\pi_{j}(\Lambda)$ for every $1\leq j\leq d$.

(iv) The trivial points of $\Lambda_{j}$ is dense in $\Lambda_{j}$ for every $1\leq j\leq d$.

Remark 1.3**.**

Zhang and Xu [35] proved that if $\Lambda$ is a slicing self-affine sponge (see Remark 3.1 for a definition), then $\Lambda$ possesses inner trivial points as soon as it possesses trivial points. This explains why in Proposition 1.2, we require that $\Lambda_{j}$ possesses trivial points instead of inner trivial points of $\Lambda_{j}$.

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Theorem 1.3**.**

Let $\Lambda$ be a non-degenerated Lalley-Gatzouras sponge, and let $\mu$ be the canonical Bernoulli measure. If $\Lambda$ does not satisfy the maximal power law, then there exists a real number $0<\chi<\dim_{B}\Lambda$ such that for any cylinder $R$,

[TABLE]

Remark 1.4**.**

We remark that if a cylinder box-counting measure $\mu$ is also a component-counting measure, then for any cylinder $R$ of $\Lambda$ and $\delta$ small enough, it holds that

[TABLE]

*which means that even locally, a large portion of $\delta$-connected components are very ‘small’.

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The paper is organized as follows. Theorem 1.1 is proved in Section 2. In Section 3, we recall some known results about Lalley-Gatzouras type sponges. In Section 4, we give some notations and lemmas. In Section 5, we deal with the case that there exists $j$ such that $\Lambda_{j}$ contains no inner trivial points. Theorem 1.3 and Theorem 1.2 are proved in Section 6.

2. Component-counting measures of self-conformal sets

Let $\alpha>0$. A conformal map $S:V\to\mathbb{R}^{d}$ is $\mathcal{C}^{1+\alpha}$ differentiable if there exists a constant $C>0$ such that

[TABLE]

The attractor of an IFS $\{S_{j}\}_{j=1}^{N}$ is called a $\mathcal{C}^{1+\alpha}$ self-conformal set, if all maps in the IFS are $\mathcal{C}^{1+\alpha}$ differentiable.

In this section, we always assume that $K$ is a $\mathcal{C}^{1+\alpha}$ self-conformal set generated by the IFS $\{S_{j}\}_{j=1}^{N}$. Let $s=\dim_{H}K(=\dim_{B}K)$ and $\mu$ be the Gibbs measure on $K$.

For $A\subset\mathbb{R}^{d}$, we denote $|A|$ the diameter of $A$. The following lemmas are well-known.

Lemma 2.1** (Principle of bounded distortion [27, 28]).**

Let $E$ be a compact subset of $V$. Then there exists $C_{1}\geq 1$ such that for any ${\boldsymbol{\sigma}}\in\Sigma^{*}$,

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for all $x,y\in E$.

Strings ${\boldsymbol{\omega}},{\boldsymbol{\tau}}\in\Sigma^{*}$ are incomparable if each is not a prefix of the other.

Lemma 2.2** (Alfors regularity [27, 28]).**

If $K$ satisfies the OSC, then $\mu\asymp{\mathcal{H}}^{s}|_{K}$, and there is a constant $C_{2}>0$ such that for any ${\boldsymbol{\sigma}}\in\Sigma^{*}$,

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Moreover, $K_{\boldsymbol{\omega}}$ and $K_{\boldsymbol{\tau}}$ are disjoint in $\mu$ provided that ${\boldsymbol{\omega}}$ and ${\boldsymbol{\tau}}$ are incomparable.

Proof of Theorem 1.1.

First, let $K$ be a self-conformal set satisfying the maximal power law. Let $K_{I}$ be a cylinder and let $\delta<|K_{I}|$. Set $\delta^{\prime}=C_{1}\delta/|K_{I}|$, where $C_{1}$ is the constant in Lemma 2.1. Let $U_{1}$ and $U_{2}$ be two $\delta^{\prime}$-connected components of $K$. Then by Lemma 2.1, $x\in S_{I}(U_{1})$ and $y\in S_{I}(U_{2})$ belongs to different $\delta$-connected component of $K_{I}$. Therefore,

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The other direction inequality can be proved in the same manner. It follows that $\mu$ is a component-counting measure.

On the other hand, if $\mu$ is a component-counting measure, then obviously $K$ satisfies the maximal power law (see Remark 1.2). The theorem is proved. ∎

3. Self-affine sponges of Lalley-Gatzouras type

In this section, we recall some known results on Lalley-Gatzouras type sponges.

3.1. Lalley-Gatzouras type sponges

We call $f:\mathbb{R}^{d}\to\mathbb{R}^{d}$, $f(x)=Tx+b$ a diagonal self-affine mapping if $T$ is a $d\times d$ diagonal matrix such that all the diagonal entries are positive numbers. An IFS $\Phi=\{\phi_{j}(x)\}_{j=1}^{m}$ is called a diagonal self-affine IFS if all the maps $\phi_{j}(x)$ are distinct diagonal self-affine contractions; the attractor is called a diagonal self-affine sponge, and we denote it by $\Lambda_{\Phi}$. Without loss of generality, we will always assume that $\Lambda_{\Phi}\subset[0,1]^{d}$. The following is an alternative definition.

Definition 3.1** (Diagonal IFS [6]).**

Let $d\geq 1$ be an integer. For each $i\in\left\{1,...,d\right\}$, let $A_{i}=\left\{0,1,...,n_{i}-1\right\}$ with $n_{i}\geq 2$, and let $\varPhi_{i}=\left(\phi_{a,i}\right)_{a\in A_{i}}$ be a collection of contracting similarities of $\left[0,1\right]$, called the base IFS in coordinate $i$. Let $A=\prod_{i=1}^{d}{A_{i}}$, and for each $a=\left(a_{1},\cdots,a_{d}\right)\in A$, consider the contracting affine maps $\phi_{a}:\left[0,1\right]^{d}\rightarrow\left[0,1\right]^{d}$ defined by the formula*

[TABLE]

where $\phi_{a,i}$ is shorthand for $\phi_{a_{i},i}$ in the formula above. Then we can get

[TABLE]

*Given $\mathcal{D}\subset A$, we call the collection $\varPhi=\left(\phi_{a}\right)_{a\in\mathcal{D}}$ a diagonal IFS, and we call its invariant set $\Lambda$ a diagonal self-affine sponge.

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Remark 3.1**.**

A diagonal self-affine IFS $\varPhi$ is called a slicing self-affine IFS, if for each $i\in\left\{1,...,d\right\}$,

[TABLE]

*is a partition of $[0,1)$ from left to right; in this case, we call $\Lambda$ a slicing self-affine sponge. In particular, if $d=2$, then we call $\Lambda$ a Barański carpet. Furthermore, if for each $1\leq j\leq d$, the contraction ratios of $\phi_{a,j}$, $a\in{\mathcal{D}}$, are all equal, then $\Lambda$ is the self-affine Sierpi$\acute{\text{n}}$ski sponge.

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We say that $\Phi$ satisfies the coordinate ordering condition if

[TABLE]

where $f^{\prime}$ denotes the derivative of the function $f$.

Recall that $\pi_{j}(x_{1},\dots,x_{d})=(x_{1},\dots,x_{j})$. Let

[TABLE]

which is an IFS on $\mathbb{R}^{j}$. Clearly $\Lambda_{j}=\pi_{j}\left(\Lambda\right)$ is the attractor of the IFS $\varPhi_{\left\{1,\cdots,j\right\}}$.

Definition 3.2** ([6]).**

Let $\Lambda_{\Phi}$ be a diagonal self-affine sponge satisfying (3.1). We say $\Phi$ satisfies the neat projection condition, if for each $j\in\{1,\dots,d\}$, the IFS $\Phi_{\{1,\dots,j\}}$ satisfies the OSC with the open set $\mathbb{I}^{j}=(0,1)^{j}$, that is,*

[TABLE]

are disjoint.* Moreover, we say a diagonal self-affine spong $\Lambda_{\Phi}$ is of Lalley-Gatzouras type if it satisfies the coordinate ordering condition as well as the neat projection condition.*

3.2. The canonical Bernoulli measure

Let ${\mathbf{p}}=(p_{\mathbf{d}})_{{\mathbf{d}}\in{\mathcal{D}}}$ be a probability weight. The unique probability measure $\mu_{\mathbf{p}}$ satisfying

[TABLE]

is called the Bernoulli measure determined by the weight ${\mathbf{p}}$.

Let $\Lambda_{\Phi}$ be a Lalley-Gatzouras type sponge. Now we define a sequence $\{\beta_{j}\}_{j=1}^{d}$ related to $\Lambda_{\Phi}$. Let $\beta_{1}>0$ be the unique real number satisfying

[TABLE]

If $\beta_{1},\dots,\beta_{j-1}$ are defined, we define $\beta_{j}>0$ to be the unique real number such that

[TABLE]

Next, for ${\mathbf{f}}=(f_{1},\dots,f_{j})\in\Phi_{\{1,\dots,j\}}$, define

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Let $\mu_{j}$ be the Bernoulli measure on $\Lambda_{j}$ defined by the weight $(p_{\mathbf{f}})_{{\mathbf{f}}\in\Phi_{\{1,\dots,j\}}}$. Especially, we denote $\mu:=\mu_{d}$, and we call $\mu$ the canonical Bernoulli measure of $\Lambda$. It is shown that

Theorem 3.1**.**

([12]) Let $\Lambda$ be a Lalley-Gatzouras type sponge, then

[TABLE]

and the canonical Bernoulli measure $\mu$ is a cylinder box-counting measure.

Denote

[TABLE]

Especially $\dim_{B}\Lambda=\alpha_{d}:=\alpha$.

Remark 3.2**.**

*Let $R$ and $\widetilde{R}$ be two $\ell$-th cylinders of $\Lambda_{\Phi}$, and $\nu$ be a Bernoulli measure of $\Lambda_{\Phi}$. Then $\nu(R\cap\widetilde{R})=0$ is always true. See for instance [31].

---

Lemma 3.3**.**

Let $\Lambda$ be a non-degenerated diagonal self-affine sponge of Lalley-Gatzouras type. Then all $\Lambda_{j}$, $1\leq j\leq d$, are non-degenerated.

Proof.

Define $\pi^{\prime}_{k}(x_{1},\dots,x_{d})=x_{k}$. Then $\Lambda$ is non-degenerated if and only if that for each $1\leq k\leq d$, $\pi^{\prime}_{k}(\Lambda)\neq\{0\}$ and $\pi^{\prime}_{k}(\Lambda)\neq\{1\}$. Since for $k\leq j$, $\pi^{\prime}_{k}\circ\pi_{j}=\pi^{\prime}_{k}$, we obtain the lemma. ∎

4. Notations and lemmas

In this section, we always assume that $\Lambda$ is a non-degenerated generalized Lalley-Gatzouras type sponge. We will use ${\mathcal{D}}$ as the alphabet and denote ${\mathcal{D}}^{*}=\bigcup_{n\geq 1}{\mathcal{D}}^{n}$. For $i_{1}\dots i_{k}\in{\mathcal{D}}^{k}$, we call $\Lambda_{i_{1}\dots i_{k}}:=\phi_{i_{1}\dots i_{k}}(\Lambda)$ a $k$-th cylinder; moreover, we set

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to be the ‘shortest side’ of $W$. Let

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Let

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and we call it the $\delta$-blocking of $\Lambda$. Clearly for $H\in{\mathcal{V}}_{\delta}$, it holds that

[TABLE]

The following lemma is obvious.

Lemma 4.1**.**

(i) For any bounded set $A\subset\mathbb{R}^{d}$, it holds that $h_{A}(\delta)\leq 3^{d}N_{A}(\delta)$.

(ii) $h_{A}(\delta)+h_{B}(\delta)\geq h_{A\cup B}(\delta)$.

(iii) $h_{A}(\delta)\geq h_{f(A)}(\delta)$ if $f$ is a contractive map.

We use $\partial E$ to denote the boundary of a set $E\subset\mathbb{R}^{d}$.

Let $W=\Lambda_{I}$ be a cylinder of $\Lambda$. We use

[TABLE]

to denote the number of $\delta$-mesh boxes intesecting $\phi_{I}(\partial[0,1]^{d}\cap\Lambda)$. The main goal of this section is to estimate $N_{W}^{b}(\delta)$. To this end, our strategy is to estimate the $\mu$-measure of the set

[TABLE]

and then use the fact that $\mu$ is a box-counting measure.

Let $F_{1},\dots,F_{2d}$ be the $(d-1)$-faces of the $[0,1]^{d}$. For $1\leq j\leq 2d$, denote

[TABLE]

Denote

[TABLE]

and set

[TABLE]

Since $\Lambda$ is non-degenerated, we conclude that for each $j$, ${\mathcal{D}}^{(j)}$ is a proper subset of ${\mathcal{D}}$ and hence $Q_{j}<1$.

Lemma 4.2**.**

For each integer $m\geq 1$, we have

[TABLE]

Proof.

Clearly $\varphi_{I}([0,1]^{d})\cap\partial F_{j}\neq\emptyset$ if and only if $I\in({\mathcal{D}}^{(j)})^{m}$, so the measure of the union of such cylinders is $Q_{j}^{m}$. The lemma is proved. ∎

Lemma 4.3**.**

Let $W$ be a cylinder of $\Lambda$ and let $m\geq 1$ be an integer. If $\delta<r_{*}^{m}S(W)$, then

[TABLE]

where $M_{0}$ is the constant in Proposition 1.3.

Proof.

Let $W=\Lambda_{I}$ be a cylinder. Since $\delta<r_{*}^{m}S(W)$, if a $\delta$-mesh box $B$ intersects $\partial W$, then $B\subset\varphi_{I}(\Lambda_{m}^{b})$. By the definition of $\mu$, we have $\mu(\varphi_{I}(A))=\mu(W)\mu(A)$ for any cylinder $A$ of $\Lambda$. Hence, by Lemma 4.2,

[TABLE]

Therefore,

[TABLE]

where the second inequality holds since $\mu$ is a box-counting measure. ∎

Finally, we characterize when $\Lambda$ admits inner trivial points.

Let $\Phi$ be an IFS with attractor $K$, and assume that $\Phi$ satisfies the strong open set condition with an open set $V$. A trivial point $x\in K$ is called an inner trivial point of $K$ if $x\in V$. Following Zhang and Huang [34], a clopen set (closed and open set) $F$ of $\Lambda$ is called a island of $\Lambda$ if $F\cap\partial\left(\left[0,1\right]^{d}\right)=\emptyset$. Obviously, an island $F$ is a union of several $k$-th cylinders for $k$ large enough.

Zhang and Huang [34] proved that if $\Phi$ is an IFS satisfying the strong open set condition, then its attractor $K$ possesses inner trivial points if and only if $K$ admits islands.

Lemma 4.4**.**

Let $\Lambda$ be a non-degenerated diagonal self-affine sponge of Lalley-Gatzouras type. Then $\Lambda$ has inner trivial points if and only if there exist $\delta>0$ and a $\delta$-connected component $X$ of $\Lambda$ such that $X\cap\partial[0,1]^{d}=\emptyset$.

Proof.

Clearly $V=(0,1)^{d}$ is an open set fulfilling the open set condition. $\Lambda$ is non-degenerated implies that $\Lambda$ satisfies the strong open set condition for this $V$. Hence, by the general result of [34], $\Lambda$ admits inner trivial points if and only if $\Lambda$ admits islands.

Suppose $\Lambda$ admits an island $U$. Let $d_{0}$ be the distance between $U$ and $\Lambda\setminus U$, and let $\delta<d_{0}/3$. Let $X$ be a $\delta$-connected component intersecting $U$. Then clearly $X$ does not intersect $\Lambda\setminus U$, and hence does not intersect $\partial[0,1]^{d}$.

On the other hand, suppose that there exists $\delta>0$ and a $\delta$-connected component $X$ of $\Lambda$ such that $X\cap\partial[0,1]^{d}=\emptyset$. Let $m$ be an integer large enough so that every $m$-th cylinder has diameter smaller than $\delta$. Then union of the $m$-th cylinders intersecting $X$ forms an island of $\Lambda$. ∎

5. The case that $\Lambda_{j}$ contains no inner trivial points for some $j$

In this section, we always assume that $\Lambda$ is a non-degenerated self-affine sponge of Lalley-Gatzouras type.

Theorem 5.1**.**

Suppose $\Lambda$ does not contain inner trivial points. Let $\tau\in(1,+\infty)$. Then there exist $0<\chi<\dim_{B}\Lambda$ and $C_{1}>0$ such that for any cylinder $W$ of $\Lambda$,

[TABLE]

Consequently, $\Lambda$ does not satisfy the maximal power law.

Proof.

Let $W=\Lambda_{I}$ be a cylinder and let $\delta<S(W)$. Let $m$ be the integer such that

[TABLE]

Since $\Lambda$ does not possess inner trivial points, by Lemma 4.4, every $\delta$-connected component of $\Lambda$ must intersect $\Lambda\cap\partial[0,1]^{d}$. It follows that every $\delta$-connected component of $W$ must intersect $W\cap\varphi_{I}(\partial[0,1]^{d})$, so

[TABLE]

By the left side inequality of (5.2), we have that for

[TABLE]

it holds that

[TABLE]

where $s=(1-\frac{1}{\tau})\frac{\log Q}{\log r_{*}}.$ Hence, the theorem holds by setting $\chi=\alpha-s$. ∎

From now on, we fix $\tau_{0}$ to be the number

[TABLE]

Then $\tau_{0}>1$ by the coordinate ordering condition; moreover, for any cylinder $W$ of $\Lambda$, we have that

[TABLE]

Lemma 5.1**.**

Suppose there exist $0<\chi<\alpha_{d-1}$ and $C_{1}>0$ such that for every cylinder $W^{\prime}$ of $\Lambda_{d-1}$, it holds that

[TABLE]

Then for every cylinder $W$ of $\Lambda$, we have

[TABLE]

where $\eta=\chi+\beta_{d}$.

Proof.

Let $H$ be a cylinder of $\Lambda$ and denote $H^{\prime}=\pi_{d-1}(H)$. We claim that for $\varepsilon<\delta/2$, it holds that

[TABLE]

Pick $x,y\in H$. If $|\pi_{d-1}(x)-\pi_{d-1}(y)|\leq\varepsilon\leq\delta/2$, then $|x-y|\leq\delta$ since $S(H)\leq\delta/2$. So if $\pi_{d-1}(x)$ and $\pi_{d-1}(y)$ belong to a same $\varepsilon$-connected component of $H^{\prime}$, then $x$ and $y$ belong to a same $\delta$-connected component of $H$, which proves our claim.

Now we turn to prove the lemma. Pick a cylinder $W$ of $\Lambda$ and let $\delta<S(W)$. Set

[TABLE]

Then for $H\in{\mathcal{V}}_{\delta^{\prime}}$, we have $\delta r_{*}/2\leq S(H)<\delta/2$. Moreover, by (5.4), we have

[TABLE]

Therefore,

[TABLE]

which confirms (5.6). The lemma is proved. ∎

Corollary 5.1**.**

Let $\Lambda$ be a non-degenerated self-affine sponge of Lalley-Gatzouras type. If there exists $j\in\{1,\dots,d\}$ such that $\Lambda_{j}$ does not contain inner trivial points, then there exists a real number $0<\chi<\dim_{B}\Lambda$ such that for any cylinder $R$,

[TABLE]

Proof.

If $j=d$, then the corollary holds by Theorem 5.1. If $j<d$, then by Lemma 5.1 and induction, we obtain (5.8). ∎

6. Proof of Theorem 1.2

In this section, we always assume that $\Lambda$ is a non-degenerated self-affine sponge of Lalley-Gatzouras type, and denote $\mu_{j}$ the canonical Bernoulli measure of $\Lambda_{j}$. Especially, $\mu:=\mu_{d}$.

Let $W=\Lambda_{I}$ be a $k$-th cylinder of $\Lambda$. We call $\phi_{I}\left(\left[0,1\right]^{d}\right)$ the corresponding $k$-th basic pillar of $W$. A $\delta$-connected component $U$ of $W$ is called an inner $\delta$-connected component, if

[TABLE]

otherwise, we call $U$ a boundary $\delta$-connected component. We denote by $h_{W}^{b}\left(\delta\right)$ the number of boundary $\delta$-connected components of $W$, and by $h_{W}^{i}\left(\delta\right)$ the number of inner $\delta$-connected components of $W$.

Lemma 6.1**.**

Then for any $\kappa>0$, there exists an integer $m_{0}\geq 1$ such that for any cylinder $W$ of $\Lambda$ and any $\delta<(r_{*})^{m_{0}}S(W)$, it holds that

[TABLE]

Proof.

Denote $\alpha=\dim_{B}\Lambda$. By Proposition 1.3, there exists $M_{0}>0$ such that for any cylinder $H$ of $\Lambda$,

[TABLE]

Let $W=\Lambda_{I}$ be a cylinder of $\Lambda$ and let $m\geq 1$. Let ${\mathcal{B}}(W,m)$ be the collection of $(k+m)$-th cylinders of $W$ whose corresponding basic pillars intersect $\varphi_{I}(\partial\left[0,1\right]^{d})$. Then

[TABLE]

where $\Lambda_{m}^{b}$ is defined by (4.4). By Lemma 4.2, we have

[TABLE]

Let $m_{0}$ be an integer such that $3^{d}M_{0}(2d)Q^{m_{0}}<\kappa$. Let $\delta<(r_{*})^{m_{0}}S(W)$; in this case if $U$ is a boundary $\delta$-connected component of $W$, then $U$ must intersect $W^{*}$. Therefore,

[TABLE]

which proves the lemma. ∎

Theorem 6.1**.**

If $\mu_{d-1}$ is a component-counting measure of $\Lambda_{d-1}$ and $\Lambda$ possesses inner trivial points. Then $\mu$ is a component-counting measure of $\Lambda$.

Proof.

Since $\Lambda$ is non-degenerated and possesses trivial points, by Lemma 4.4, there is a clopen subset $F$ of $\Lambda$ such that $F\cap\partial[0,1]^{d}=\emptyset$. Assume that $F$ is the union of $L$ number of $m_{1}$-th cylinders. Let $U$ be a $m_{1}$-cylinder in $F$ with the maximal measure in $\mu$, then

[TABLE]

Let $W$ be a $k$-th cylinder of $\Lambda$ and denote $W^{\prime}=\pi_{d-1}\left(W\right)$. Let $\delta\leq S(W)$. Since

[TABLE]

we need only to consider the lower bound estimate of $h_{W}(\delta)$.

That $\mu_{d-1}$ is a component-counting measure implies that there is a constant $M$ such that

[TABLE]

(We remark that we set the threshold $\delta_{0}(W^{\prime})=S(W^{\prime})$ here.)

Let $\kappa=M/2$ and let $m_{0}$ be the constant in Lemma 6.1. Set $\delta^{\prime}=\delta/(r_{*})^{m_{0}+m_{1}}$. Let $\mathcal{V}_{\delta^{\prime}}$ be the $\delta^{\prime}$-blocking of $\Lambda$.

Pick $H\in\mathcal{V}_{\delta^{\prime}}$ and $H\subset W$. Let $f_{H}$ be the affine map such that $H=f_{H}(\Lambda)$, then $H_{F}:=f_{H}(F)$ is an island of $H$ in the sense that $f_{H}(F)$ is a clopen subset of $H$, and $f_{H}(F)$ does not intersect $f_{H}(\partial[0,1]^{d})$. Clearly

[TABLE]

Now we estimate $h_{H_{F}}(\delta)$. Denote

[TABLE]

First, since $\pi_{d-1}$ is contractive, by Lemma 4.1(iii), we have

[TABLE]

Notice that

[TABLE]

On one hand, since $\mu_{d-1}$ is a component-counting measure, we have

[TABLE]

On the other hand, by Lemma 6.1, we have

[TABLE]

This together with (6.1) imply that

[TABLE]

First, we show that $h_{H_{F}}(\delta)$ is no less than the left hand side of (6.2):

[TABLE]

Secondly, we show that $\mu(H)$ is comparable with the right hand side of (6.2). Since

[TABLE]

we have

[TABLE]

Therefore,

[TABLE]

This proves that $\mu$ is a component-counting measure and the threshold $\delta_{0}(W)$ can be set to be $S(W)$. ∎

Proof of Theorem 1.2.

The second assertion in the theorem is proved in Corollary 5.1. In the following, we show that (i)$\Rightarrow$(ii) $\Rightarrow$ (iii)$\Rightarrow$ (i).

(i)$\Rightarrow$(ii) is trivial, see Remark 1.2.

(ii)$\Rightarrow$(iii) holds by Corollary 5.1.

(iii)$\Rightarrow$(i): First, $\mu_{1}$ is a component-counting measure of $\Lambda_{1}=\pi_{1}(\Lambda)$ by Theorem 1.1. Then by Theorem 6.1 and induction, we conclude that $\mu_{j}$ is a component-counting measure of $\Lambda_{j}$ for each $1\leq j\leq d$.

That (iii)$\Leftrightarrow(iv)$ is obvious. The theorem is proved. ∎

Proof of Theorem 1.3.

By Theorem 1.2, if $\Lambda$ does not satisfy the maximal power law, then there exists $1\leq j\leq d$ such that $\Lambda_{j}$ contains no inner trivial points. Now the theorem is a direct consequence of Corollary 5.1. ∎