Hausdorff Dimension and Topological Entropies of a Solenoid

Abstract

The purpose of this paper is to elucidate the interrelations between three essentially different concepts: dynamical solenoids, topological entropy, and Hausdorff dimension, where by a dynamical solenoid we mean a sequence of continuous epimorphisms of a compact metric space. For this purpose, we describe a dynamical solenoid by topological entropy-like quantities and investigate the relations between them. For L-Lipschitz dynamical solenoids and locally $\lambda -$expanding dynamical solenoids, we show that the topological entropy and fractal dimensions are closely related. For a locally $\lambda -$expanding dynamical solenoid, we prove that its topological entropy is lower estimated by the Hausdorff dimension of X multiplied by the logarithm of $\lambda .$

1. Introduction

A solenoid, which was introduced to mathematics by Vietoris [1] as the topological object, can be presented either in an abstract way as an inverse limit or in a geometric way as a nested intersection of a sequence of tori. The classical construction of Vietoris was modified by McCord [2], Williams [3], and others. Since the publication of William’s paper on expanding attractors [3], inverse limit spaces have played a key role in dynamical systems and in continuum theory. Smale [4] introduced the solenoid to dynamical systems as a hyperbolic attractor.

In the paper, a sequence $f_{\infty }={(f_n:X\to X)}_{n=1}^{\infty }$ of continuous epimorphisms of a compact metric space $(X,d)$ is called a dynamical solenoid while the inverse limit

$$X_{\infty }=\underset{⟵}{lim}(X,f_k):=\{{\left(x_k\right)}_{k=0}^{\infty }:x_{k-1}=f_k\left(x_k\right)\}.$$

is called a classical solenoid. Since the paper is not about classical solenoids, the term dynamical solenoid is sometimes abbreviated as solenoid.

In mathematical literature, one can also find a more restrictive definition of the solenoid as a finite-dimensional, connected, compact abelian group. These solenoids generalize torus groups, and their entropic properties have been studied by Berg [5], Lind and Ward [6], Einsiedler and Lindenstrauss [7], and others. A less restrictive definition of the solenoid was considered in [8,9,10].

A dynamical solenoid is a natural generalization of a classical dynamical system. In contrast with the classical dynamical systems, the properties of solenoid entropies have not been fully investigated. In the paper, we consider several different definitions of entropy-like quantities for a dynamical solenoid $f_{\infty }$: topological entropy $h_{top}\left(f_{\infty }\right)$, topological cover entropy $h_{top-cov}\left(f_{\infty }\right)$, and topological dimensional entropy $h_{top-dim}\left(f_{\infty }\right).$

Both nonautonomous dynamical systems and dynamical solenoids are determined by compositions of continuous self-maps; therefore, in both cases, the entropy-like quantities that capture complexities of these generalized dynamical systems can be similar. For example, the topological entropy of a dynamical solenoid coincides with the topological entropy of a nonautonomous dynamical system defined in [11]. In this paper, we derive the following relations between the entropies of a dynamical solenoid which were previously known for continuous maps on compact metric spaces, and we obtained the following results.

Theorem 1.

$h_{top-dim}\left(f_{\infty }\right)\le h_{top-cov}\left(f_{\infty }\right).$

Theorem 2.

$h_{top}\left(f_{\infty }\right)=h_{top-cov}\left(f_{\infty }\right).$

In 2002, Milnor [12] stated two questions related to the classical dynamical system: “Is entropy of it effectively computable?” “Given an explicit dynamical system and given $ϵ>0,$ is it possible to compute the entropy with maximal error of $ϵ$?” In most cases the answer is negative. For the recent results on computability of topological entropy, we recommend [13,14].

Therefore, in mathematical literature, there were many attempts to estimate entropy of dynamical systems by Lyapunov exponents, volume growth, Hausdorff dimension, or fractal dimensions.

The theory of Carathéodory structures, introduced by Pesin [15] for a single map, has been applied in [11] to get some estimations of the topological entropy of a nonautonomous dynamical system. To show a comprehensive picture and beauty of dynamics of dynamical solenoids, we rewrite the Theorem 3 in [11] to express complexity of so called L-Lipschitz dynamical solenoid. A dynamical solenoid $f_{\infty }={(f_n:X\to X)}_{n=1}^{\infty }$ is called L-Lipschitz if it consists of L-Lipschitz epimorphisms; the following inequality holds.

Theorem 3.

Assume that $f_{\infty }={(f_n:X\to X)}_{n=1}^{\infty }$ is a L-Lipschitz dynamical solenoid with $L>1$. Then, for any $Y\subset X$, we obtain

$$HD\left(Y\right)\ge \frac{h_{top-dim}(\left(f_{\infty }\right),Y)}{log\left(L\right)},$$

where $HD\left(Y\right)$ is the Hausdorff dimension of $Y.$

Finally, we investigate so called locally $\lambda -$expanding dynamical solenoids, in the sense of Ruelle [16] (see Definition 6). We prove that the topological entropy of a $\lambda -$expanding dynamical solenoid, defined on the space $X,$ is related to the upper box dimension of X multiplied by the logarithm of $\lambda .$ We obtained the following inequalities.

Theorem 4.

Given a locally $\lambda -$expanding dynamical solenoid $f_{\infty }={(f_n:X\to X)}_{n=1}^{\infty }$. Then,

$$h_{top}\left(f_{\infty }\right)\ge (log\lambda )·\overline{{dim}_B\left(X\right)}\ge (log\lambda )·HD\left(X\right),$$

where $\overline{{dim}_B\left(X\right)}$ is the upper box dimension of $X.$

The paper is organized as follows. In Section 2, we introduce several definitions of entropy-like quantities for a dynamical solenoid: topological entropy, topological cover entropy, and topological dimensional entropy. In Section 3, we prove the relations between them (Theorems 1 and 2). Section 4 is devoted to L-Lipschitz dynamical solenoids; we present Theorem 3. Finally, in Section 5, we investigate locally $\lambda -$expanding dynamical solenoids and prove Theorem 4.

2. Topological Entropies of a Dynamical Solenoid

In 1965, Adler, Konheim, and McAndrew [17] introduced a definition of topological entropy for the classical dynamical system (i.e., a pair $(X,f),$ where X is a topological space and $f:X\to X$ is a continuous map) as a non-negative number assigned to an open cover of $X.$ A different definition of entropy of a continuous self-map defined on a compact metric space was introduced by Bowen [18] and independently by Dinaburg [19]. In [20], Bowen proved that the definitions are equivalent. Nowadays, topological entropy is a main notion in topological dynamics. In the paper, we present a few generalizations of the classical topological entropy of a single map to dynamical solenoids.

In the paper, we consider a dynamical solenoid determined by a sequence $f_{\infty }={(f_n:X\to X)}_{n=1}^{\infty }$ of continuous epimorphisms of a compact metric spaces $(X,d).$ Thus, the dynamical solenoid is a generalized dynamical system. Its complexity and chaos can be measured by several entropy-like quantities. First, we introduce topological entropy via $(n,ϵ)-$separated sets.

2.1. Topological Entropy of a Dynamical Solenoid via $(n,ϵ)-$Separated or $(n,ϵ)-$Spanning Sets

Let $B(x,r)=\{y\in X:d(x,y)\le r\}$ denote a closed ball in the metric space $(X,d)$ centered at $x\in X$ and with radius $r.$

Definition 1.

Fix $ϵ>0$, $n\in \mathbb{N}.$ A subset $F\subset X$ is called $(n,ϵ)$-spanning if for any $x\in X$ there exists $y\in F$ such that

$$max\{d(f_i\circ f_{i+1}\circ \dots \circ f_n\left(x\right),f_i\circ f_{i+1}\circ \dots \circ f_n\left(y\right)):i\in \{1,\dots ,n\}\}\le ϵ.$$

Let $r(n,ϵ):=min\left\{card\right(F):Fis(n,ϵ)$-$spanningset\}.$

A set $E\subset X$ is called $(n,ϵ)$-separated if for any pair of distinct points $x,y\in E$ we have

$$max\{d(f_i\circ f_{i+1}\circ \dots \circ f_n\left(x\right),f_i\circ f_{i+1}\circ \dots \circ f_n\left(y\right)):i\in \{1,\dots ,n\}\}>ϵ.$$

Let $s(n,ϵ):=max\left\{card\right(E):Eis(n,ϵ)$-$separated\}.$

The following two lemmas are a reformulation of Definition 1.

Lemma 1.

A set $F\subset X$ is $(n,ϵ)-$spanning if and only if

$$X=\bigcup _{y\in F}\bigcap _{i=1}^n{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}B[(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(y\right),ϵ].$$

Proof. 

$(\Rightarrow )$ Assume that a subset $F\subset X$ is $(n,ϵ)-$spanning. Then for any point $x\in X$ there exists a point $y\in F$ such that

$$max\{d(f_i\circ f_{i+1}\circ \dots \circ f_n\left(x\right),f_i\circ f_{i+1}\circ \dots \circ f_n\left(y\right)):i\in \{1,\dots ,n\}\}\le ϵ.$$

For any $i\in \{1,\dots ,n\}$ we obtain

$$f_i\circ f_{i+1}\circ \dots \circ f_n\left(x\right)\in B[f_i\circ f_{i+1}\circ \dots \circ f_n\left(y\right),ϵ]$$

and

$$x\in {(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}B[(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(y\right),ϵ].$$

So

$$x\in \bigcap _{i=1}^n{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}B[(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(y\right),ϵ].$$

Since x is an arbitrary point of X we conclude

$$X\subset \bigcup _{y\in F}\bigcap _{i=1}^n{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}B[(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(y\right),ϵ]\subset X.$$

$(\Leftarrow )$ Now assume that the following equality

$$X=\bigcup _{y\in F}\bigcap _{i=1}^n{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}B[(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(y\right),ϵ]$$

holds for a subset $F\subset X.$ Then, for any $x\in X$ there exists $y\in F$ such that

$$x\in \bigcap _{i=1}^n{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}B[(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(y\right),ϵ]$$

which is equivalent to

$$max\{d(f_i\circ f_{i+1}\circ \dots \circ f_n\left(x\right),f_i\circ f_{i+1}\circ \dots \circ f_n\left(y\right)):i\in \{1,\dots ,n\}\}\le ϵ.$$

Thus the set F is $(n,ϵ)-$spanning and the proof is finished. □

Lemma 2.

A set $E\subset X$ is $(n,ϵ)$-separated if and only if for any $x\in E$ the set ${\bigcap }_{i=1}^n{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}B[(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(x\right),ϵ]$ contains no other points of $E.$

Proof. 

$(\Rightarrow )$ Assume that a set $E\subset X$ is $(n,ϵ)-$separated and choose two distinct points $x_1,x_2\in E.$ For any $i\in \{1,\dots ,n\}$ we get

$$x_1\in {(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}B[(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(x_1\right),ϵ]$$

so

$$x_1\in \bigcap _{i=1}^n{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}B[(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(x_1\right),ϵ].$$

Assume that

$$x_2\in \bigcap _{i=1}^n{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}B[(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(x_1\right),ϵ]$$

then we obtain the following inequality

$$max\{d(f_i\circ f_{i+1}\circ \dots \circ f_n\left(x_1\right),f_i\circ f_{i+1}\circ \dots \circ f_n\left(x_2\right)):i\in \{1,\dots ,n\}\}\le ϵ$$

which gives a contradiction with the assumption that $x_1,x_2$ are $(n,ϵ)-$separated. Thus the intersection

$$(\ast )E\cap \bigcap _{i=1}^n{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}B[(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(x_1\right),ϵ]=\left\{x_1\right\}.$$

$(\Leftarrow )$ Now assume that for a given subset $E\subset X$ the condition $(\ast )$ holds. For two distinct points $x_1,x_2\in E$ we have

$$x_2\notin \bigcap _{i=1}^n{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}B[(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(x_1\right),ϵ].$$

Therefore, there exists $i\in \{1,\dots ,n\}$ such that

$$d(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(x_1\right),f_i\circ f_{i+1}\circ \dots \circ f_n\left)\left(x_2\right)\right)>ϵ.$$

We have proved that the set $E\subset X$ is $(n,ϵ)-$separated. □

Modifying slightly the classical Bowen’s definition [18] of the topological entropy of a single map (for details see also Chapter 7 in [21]), we present the definition of topological entropy of a dynamical solenoid as follows.

Definition 2.

The quantity

$$h_{top}\left(f_{\infty }\right):=\underset{ϵ\to 0^{+}}{lim}\underset{n\to \infty }{lim\; sup}\frac{1}{n}logs(n,ϵ)$$

is called the topological entropy of $f_{\infty }$.

Remark 1.

The topological entropy of a dynamical solenoid can also be expressed in the language of $(n,ϵ)$-spannings sets. Using arguments similar to remarks on page 169 in [21], we get estimations

$$r(n,ϵ)\le s(n,ϵ)\le r(n,ϵ/2).$$

Indeed, for any two distinct points $x_1,x_2$ of an $(n,ϵ)-$separated set E with cardinality $card\left(E\right)=s(n,ϵ)$ we have

$$max\{d(f_i\circ f_{i+1}\circ \dots \circ f_n\left(x_1\right),f_i\circ f_{i+1}\circ \dots \circ f_n\left(x_2\right)):i\in \{1,\dots ,n\}\}>ϵ.$$

Since E is $(n,ϵ)-$separated set with maximal cardinality, for any $y\in X\backslash E$ there exists $x_3\in E$ such that

$$max\{d(f_i\circ f_{i+1}\circ \dots \circ f_n\left(y\right),f_i\circ f_{i+1}\circ \dots \circ f_n\left(x_3\right)):i\in \{1,\dots ,n\}\}\le ϵ.$$

It means that E is $(n,ϵ)-$spanning and

$$r(n,ϵ)\le card\left(E\right)=s(n,ϵ).$$

To show the other inequality for the set E and an $(n,\frac{ϵ}{2})-$ spanning set $F\subset X$ with cardinality $card\left(F\right)=r(n,ϵ/2)$ define $\varphi :E\to F$ by choosing, for each point $x\in E$ some point $\varphi \left(x\right)\in E$ with

$$max\{d(f_i\circ f_{i+1}\circ \dots \circ f_n\left(x\right),f_i\circ f_{i+1}\circ \dots \circ f_n\left(\varphi \left(x\right)\right)):i\in \{1,\dots ,n\}\}\le \frac{ϵ}{2}.$$

The map $\varphi :E\to F$ is injective and therefore $card\left(E\right)\le card\left(F\right).$ Hence $s(n,ϵ)\le r(n,ϵ/2).$

Applying the inequalities

$$r(n,ϵ)\le s(n,ϵ)\le r(n,ϵ/2)$$

and passing to the suitable limits, we obtain the equality

$$h_{top}\left(f_{\infty }\right)=\underset{ϵ\to 0^{+}}{lim}\underset{n\to \infty }{lim\; sup}\frac{1}{n}logr(n,ϵ).$$

Remark 2.

Assume that all maps of the sequence $f_{\infty }={(f_n:X\to X)}_{n=1}^{\infty }$ coincide with a fixed continuous map $f:X\to X$ of a compact metric space $(X,d).$ Then, the topological entropy of $f_{\infty }$ is equal to the topological entropy of $f.$ For example, the topological entropy of a dynamical solenoid coincides with the topological entropy of a nonautonomous dynamical system defined in [11].

2.2. Topological Entropy of a Dynamical Solenoid via Open Covers

It is a well-known fact that topological entropy of a single continuous map $f:X\to X$ can be defined by open covers of the compact metric space $(X,d).$ We intend to show that similar approach can be applied to a dynamical solenoid. For this purpose, notice that for two open covers $\mathcal{A},$$\mathcal{B}$ of $X,$ the family

$$\mathcal{A}\vee \mathcal{B}:=\{A\cap B:A\in \mathcal{A},B\in \mathcal{B}\}$$

is an open cover of $X.$ Moreover, for a continuous map $f_i\circ f_{i+1}\circ \dots \circ f_n:X\to X$ and an open cover $\mathcal{A}$ of X the family

$${(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}\mathcal{A}:=\{{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}A:A\in \mathcal{A}\}$$

is an open cover of $X.$ Thus, for the open cover $\mathcal{A}$ of $X,$ the family

$$\underset{i=1}{\overset{n}{\bigvee }}{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}\mathcal{A}:=$$

$${(f_1\circ f_2\circ \dots \circ f_n)}^{-1}\left(\mathcal{A}\right)\vee {(f_2\circ f_3\circ \dots \circ f_n)}^{-1}\left(\mathcal{A}\right)\vee \dots \vee {\left(f_n\right)}^{-1}\left(\mathcal{A}\right)$$

is an open cover of $X.$

For an open cover $\mathcal{B}$ of X let us denote by $N\left(\mathcal{B}\right)$ the number of sets in a finite subcover of $\mathcal{B}$ covering $X,$ with the smallest cardinality.

Definition 3.

The topological cover entropy of $f_{\infty },$ relative to an open cover $\mathcal{A}$ of $X,$ is defined as

$$h_{top-cov}(f_{\infty },\mathcal{A}):=\underset{n\to \infty }{lim\; sup}\frac{1}{n}logN\left(\underset{i=1}{\overset{n}{\bigvee }}{(f_i\circ f_{i+1}\circ ...\circ f_n)}^{-1}\mathcal{A}\right),$$

whereas the topological cover entropy of $f_{\infty }$ is the quantity

$$h_{top-cov}\left(f_{\infty }\right):=\underset{\mathcal{A}}{sup}h(f_{\infty },\mathcal{A}),$$

where $\mathcal{A}$ ranges over all open covers of X.

2.3. Topological Entropy as a Dimension Theory Quantity

Here, we modify the Bowen’s definition [20] of the topological entropy of a continuous single map, which is similar to the construction of the Hausdorff measure, to obtain the topological dimensional entropy of $f_{\infty }$.

2.3.1. The Hausforff Measure and the Hausdorff Dimension

For the convenience of the reader, we recall briefly the classical construction of the Hausdorff measure and the Hausdorff dimension.

For a metric space $(X,d)$ and a subset $Y\subset X$, let us denote by $Co{v}_{ϵ}\left(Y\right)$ the family of open covers $\mathcal{B}$ of Y with $diam\left(B\right)<ϵ,$ for any $B\in \mathcal{B}.$ Here, $diam\left(B\right)$ denotes the diameter of $B.$

For any $\lambda >0$ the classical Hausdorff $\lambda -$measure ${\mu }_{\lambda }\left(Y\right)$ of a subset $Y\subset X$ is defined as follows,

$${\mu }_{\lambda }\left(Y\right):=\underset{ϵ\to 0}{lim}inf\{\sum _{B\in \mathcal{B}}{\left[diam\left(B\right)\right]}^{\lambda }:\mathcal{B}\in Co{v}_{ϵ}\left(Y\right)\}.$$

The function $\lambda \to {\mu }_{\lambda }\left(Y\right)$ has a unique critical point, where it jumps from ∞ to 0. The Hausdorff dimension $HD\left(Y\right)$ of Y is defined as the critical point of the function $\lambda \to {\mu }_{\lambda }\left(Y\right),$ i.e.,

$$HD\left(Y\right)=sup\{\lambda :{\mu }_{\lambda }\left(Y\right)=\infty \}=inf\{\lambda :{\mu }_{\lambda }\left(Y\right)=0\}.$$

2.3.2. Generalized Hausdorff Measure and Generalized Hausdoff Dimension

Arguments similar to the construction of the classical Hausdorff $\lambda$-measure and the Hausdorff dimension lead to another entropy-like quantity for $f_{\infty }={(f_n:X\to X)}_{n=1}^{\infty }$. Denote by $\mathcal{A}$ a finite open cover of $X.$ For a subset $B\subset X,$ we write $B\prec \mathcal{A}$ if there exists $A_i\in \mathcal{A}$ such that $B\subset A_i.$ Denote by $n_{\mathcal{A}}\left(B\right)$ the largest non-negative integer n such that $f_k\circ f_{k+1}\circ \dots \circ f_n\left(B\right)\prec \mathcal{A}$ for $k=1,\dots ,n.$ If there is no element $A_i\in \mathcal{A}$ such that $B\subset A_i,$ then we write $n_{\mathcal{A}}\left(B\right)=0.$ Let us introduce the following notations.

$$dia{m}_{\mathcal{A}}\left(B\right):=exp(-n_{\mathcal{A}}\left(B\right)),$$

$$dia{m}_{\mathcal{A}}\left(\mathcal{B}\right):=sup\{dia{m}_{\mathcal{A}}\left(B\right):B\in \mathcal{B}\}$$

and

$${\mathcal{D}}_{\mathcal{A}}(\mathcal{B},\lambda ):=\sum _{B\in \mathcal{B}}{\left[dia{m}_{\mathcal{A}}\left(B\right)\right]}^{\lambda }$$

for a family $\mathcal{B}$ of subsets of X and a real number, $\lambda >0.$ For a subset $Y\subset X$ and $ϵ>0$, let $Co{v}_{ϵ}^{\mathcal{A}}\left(Y\right)$ denote the family of open covers $\mathcal{B}$ of Y with $dia{m}_{\mathcal{A}}\left(\mathcal{B}\right)<ϵ.$ Now we set

$${\mu }_{\mathcal{A},\lambda }\left(Y\right):=\underset{ϵ\to 0}{lim}inf\{{\mathcal{D}}_{\mathcal{A}}(\mathcal{B},\lambda ):\mathcal{B}\in Co{v}_{ϵ}^{\mathcal{A}}\left(Y\right)\}.$$

The behavior of the function $\lambda \to {\mu }_{\mathcal{A},\lambda }\left(Y\right)$ is very similar to the behavior of $\lambda \to {\mu }_{\lambda }\left(Y\right)$: it has a unique critical point, where it jumps from ∞ to 0. More precisely.

Lemma 3.

For the function $\lambda \to {\mu }_{\mathcal{A},\lambda }\left(Y\right)$, there exists a unique critical number ${\lambda }_0$ such that ${\mu }_{\mathcal{A},\lambda }\left(Y\right)=\infty ,for0\le \lambda <{\lambda }_{0}and$ ${\mu }_{\mathcal{A},\lambda }\left(Y\right)=0,for{\lambda }_0<\lambda .$

Proof. 

For any $ϵ\in (0,1)$ there exists a cover $\mathcal{B}$ of Y with $exp(-n_{\mathcal{A}}\left(B\right))<1,$ for any $B\in \mathcal{B}.$ Therefore, the inequality $0<\beta <\alpha$ implies

$$\sum _{B\in \mathcal{B}}exp(-n_{\mathcal{A}}\left(B\right)·\alpha )\le \sum _{B\in \mathcal{B}}exp(-n_{\mathcal{A}}\left(B\right)·\beta ),$$

so

$${\mu }_{\mathcal{A},\alpha }\left(Y\right)=\underset{ϵ\to 0}{lim}inf\{\sum _{B\in \mathcal{B}}exp(-n_{\mathcal{A}}\left(B\right)·\alpha ):\mathcal{B}\in Co{v}_{ϵ}^{\mathcal{A}}\left(Y\right)\}\le$$

$$\underset{ϵ\to 0}{lim}inf\{\sum _{B\in \mathcal{B}}exp(-n_{\mathcal{A}}\left(B\right)·\beta ):\mathcal{B}\in Co{v}_{ϵ}^{\mathcal{A}}\left(Y\right)\}={\mu }_{\mathcal{A},\beta }\left(Y\right).$$

Therefore,

$$(\ast \ast )0<\beta <\alpha \Rightarrow {\mu }_{\mathcal{A},\alpha }\left(Y\right)\le {\mu }_{\mathcal{A},\beta }\left(Y\right).$$

First assume that ${\mu }_{\mathcal{A},\delta }\left(Y\right)=\infty$ for some $\delta >0$ and that $\beta <\delta .$ By $(\ast \ast )$ we conclude that

$$\infty ={\mu }_{\mathcal{A},\delta }\left(Y\right)\le {\mu }_{\mathcal{A},\beta }\left(Y\right).$$

In a similar way, we prove that if ${\mu }_{\mathcal{A},\lambda }\left(Y\right)=0,$ then for ${\lambda }_1>\lambda$ we obtain the equality ${\mu }_{\mathcal{A},{\lambda }_1}\left(Y\right)=0.$ □

Definition 4.

Denote by ${\lambda }_0$ the critical point of the function $\lambda \to {\mu }_{\mathcal{A},\lambda }\left(Y\right).$ Let ${\lambda }_0=h_{top-dim}(\left(f_{\infty }\right),Y,\mathcal{A}).$ In other words, let

$$h_{top-dim}(\left(f_{\infty }\right),Y,\mathcal{A}):=sup\{\lambda :{\mu }_{\mathcal{A},\lambda }\left(Y\right)=\infty \}=inf\{\lambda :{\mu }_{\mathcal{A},\lambda }\left(Y\right)=0\}.$$

The number

$$h_{top-dim}(f_{\infty },Y):=sup\{h_{top-dim}(\left(f_{\infty }\right),Y,\mathcal{A}):\mathcal{A}finiteopencoverofY\}$$

is called the topological dimensional entropy of $f_{\infty }$ restricted to $Y.$ If $Y=X$, we write $h_{top-dim}(f_{\infty },X)=h_{top-dim}\left(f_{\infty }\right).$

Remark 3.

Our definition of topological dimension entropy of a dynamical solenoid is an extension of Bowen’s entropy [20]. Moreover, the topological dimensional entropy of a dynamical solenoid is similar to Bowen’s topological entropy of nonautonomous dynamical systems in [22].

3. Relations between Topological Entropies of a Dynamical Solenoid

In the previous section, we introduced three entropy-like quantities for a dynamical solenoid. Now, we relate the topological dimensional entropy of a dynamical solenoid to its topological covering entropy. We obtain the following result.

Theorem 1.

$h_{top-dim}\left(f_{\infty }\right)\le h_{top-cov}\left(f_{\infty }\right).$

Proof. 

Choose a finite open cover $\mathcal{A}$ of X and let

$${\mathcal{A}}_n=\{\bigcap _{i=1}^n{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}\left(A_i\right):A_i\in \mathcal{A}\}.$$

Denote by ${\mathcal{B}}_n$ a finite subcover of ${\mathcal{A}}_n$ with cardinality $|{\mathcal{B}}_n|=N\left({\mathcal{A}}_n\right).$ Then, for any $B\in {\mathcal{B}}_n,$ we obtain that $n_{\mathcal{A}}\left(B\right)\ge n,$ so

$$dia{m}_{\mathcal{A}}\left(B\right)\le exp(-n)$$

and for any $\lambda >0$ we get

$${\mathcal{D}}_{\mathcal{A}}({\mathcal{B}}_n,\lambda )=\sum _{B\in {\mathcal{B}}_n}{\left[dia{m}_{\mathcal{A}}\left(B\right)\right]}^{\lambda }=\sum _{B\in {\mathcal{B}}_n}exp(-\lambda ·n_{\mathcal{A}}\left(B\right))\le \left|{\mathcal{B}}_n\right|·exp(-\lambda ·n).$$

As $|{\mathcal{B}}_n|=N\left({\mathcal{A}}_n\right),$ we have

$$|{\mathcal{B}}_n|·exp(-\lambda ·n)=exp(-\lambda ·n+log|{\mathcal{B}}_n\left|\right)=exp(-n(\lambda -\frac{1}{n}logN\left({\mathcal{A}}_n\right))).$$

Consequently,

$${\mathcal{D}}_{\mathcal{A}}({\mathcal{B}}_n,\lambda )\le exp(-n·(\lambda -\frac{1}{n}logN\left({\mathcal{A}}_n\right))).$$

Fix $ϵ>0$ and an arbitrary small $\gamma >0.$ Choose ${\lambda }^{\ast }$ such that ${\lambda }^{\ast }>h_{top-cov}(f_{\infty },\mathcal{A})>{\lambda }^{\ast }-\gamma .$ For sufficiently large $n\in \mathbb{N}$, we obtain the inequalities

$${\lambda }^{\ast }-\frac{1}{n}logN\left({\mathcal{A}}_n\right))>0,$$

$$dia{m}_{\mathcal{A}}\left(B\right)<exp(-n)<ϵ,forB\in \mathcal{B},and$$

$${\mathcal{D}}_{\mathcal{A}}({\mathcal{B}}_n,{\lambda }^{\ast })\le exp(-n·({\lambda }^{\ast }-\frac{1}{n}logN\left({\mathcal{A}}_n\right)))<ϵ.$$

As $ϵ>0$ is arbitrarily small, the above two inequalities yield ${\mu }_{\mathcal{A},{\lambda }^{\ast }}\left(X\right)=0.$ Therefore,

$$h_{top-dim}(f_{\infty },Y,\mathcal{A})\le {\lambda }^{\ast }\le h_{top-cov}(f_{\infty },\mathcal{A})+\gamma .$$

As $\mathcal{A}$ is an arbitrary finite open cover of $X,$ we obtain

$$h_{top-dim}\left(f_{\infty }\right)=sup\{h_{top-dim}(\left(f_{\infty }\right),X,\mathcal{A}):\mathcal{A}-finiteopencoverofX\}$$

$$\le sup\{h_{top-cov}(f_{\infty },\mathcal{A}):\mathcal{A}-finiteopencoverofX\}+\gamma =h_{top-cov}\left(f_{\infty }\right)+\gamma .$$

Finally, passing with $\gamma$ to zero, we get

$$h_{top-dim}\left(f_{\infty }\right)\le h_{top-cov}\left(f_{\infty }\right).$$

 □

Lemma 4.

For an open cover $\mathcal{A}$ of X with the Lebegue number $Leb\left(\mathcal{A}\right)=\delta ,$ we get

$$N\left(\underset{i=1}{\overset{n}{\bigvee }}{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}\mathcal{A}\right)\le r(n,\frac{\delta }{2}).$$

Proof. 

Fix $n\in \mathbb{N}$ and $\delta >0.$ Choose an $(n,\frac{\delta }{2})$-spanning set F with cardinality $card\left(F\right)=r(n,\frac{\delta }{2}).$ As $Leb\left(\mathcal{A}\right)=\delta$, we obtain that any ball $B[(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(x\right),\frac{\delta }{2}]$ of radius $\delta /2,$ where $x\in F$ and $i\in \{1,\dots ,n\},$ is included in some set $A_i\in \mathcal{A},$ so

$$\bigcap _{i=1}^n{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}B[(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(x\right),\frac{\delta }{2}]\subset \bigcap _{i=1}^n{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}{A}_i,$$

for some $A_1,A_2,\dots A_n\in \mathcal{A}.$ It means that the set

$$\bigcap _{i=1}^n{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}B[(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(x\right),\frac{\delta }{2}]$$

is a subset of some member of the covering

$$\underset{i=1}{\overset{n}{\bigvee }}{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}\mathcal{A}.$$

On the other hand, applying Lemma 1, we get

$$X=\bigcup _{x\in F}\bigcap _{i=1}^n{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}B((f_i\circ f_{i+1}\circ \dots \circ f_n)\left(x\right),\frac{\delta }{2}),$$

so

$$N\left(\underset{i=1}{\overset{n}{\bigvee }}{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}\mathcal{A}\right)\le card\left(F\right)=r(n,\frac{\delta }{2}).$$

 □

Lemma 5.

Assume that $ϵ>0$ and $\mathcal{B}$ is an open cover of $X,$ with $diam\left(\mathcal{B}\right)\le ϵ.$ Then,

$$s(n,ϵ)\le N\left(\underset{i=1}{\overset{n}{\bigvee }}{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}\mathcal{B}\right).$$

Proof. 

Choose an $(n,ϵ)$-separated set E with cardinality $card\left(E\right)=s(n,ϵ).$ Assume that two distinct points $x_1,x_2\in E$ belong to the same member of the cover ${\bigvee }_{i=1}^n{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}\mathcal{B}.$ Therefore, there exist sets $B_i\in \mathcal{B}$ such that $(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(x_1\right),(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(x_2\right)\in B_i$ for any $i\in \{1,..,n\}.$ On the other hand, as the set E is $(n,ϵ)$-separated, there exists $j\in \{1,\dots ,n\}$ such that

$$d((f_j\circ f_{j+1}\circ \dots \circ f_n)\left(x_1\right),(f_j\circ f_{j+1}\circ \dots \circ f_n)\left(x_2\right))=$$

$$max\{d(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(x_1\right),(f_i\circ f_{i+1}\circ \dots \circ f_n)\left(x_2\right):i\in \{1,..,n\}\}>ϵ.$$

Thus, we get a contradiction with $diam\left(B_j\right)\le ϵ.$ Therefore,

$$s(n,ϵ)\le N\left(\underset{i=1}{\overset{n}{\bigvee }}{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}\mathcal{B}\right).$$

 □

Now, we are ready to prove that the topological entropy of a dynamical solenoid is equivalent to its topological covering entropy.

Theorem 2.

$h_{top}\left(f_{\infty }\right)=h_{top-cov}\left(f_{\infty }\right).$

Proof. 

Fix $ϵ>0.$ Let ${\mathcal{A}}_{ϵ}$ be the cover of X by all open balls of radius $2·ϵ$ and denote by ${\mathcal{B}}_{ϵ}$ the cover of X by all open balls of radius $\frac{ϵ}{2}.$ Due to Lemma 4, we obtain

$$N\left(\underset{i=1}{\overset{n}{\bigvee }}{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}{\mathcal{A}}_{ϵ}\right)\le r(n,ϵ),$$

so

$$\underset{n\to \infty }{lim\; sup}\frac{1}{n}logN\left(\underset{i=1}{\overset{n}{\bigvee }}{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}{\mathcal{A}}_{ϵ}\right)\le \underset{n\to \infty }{lim\; sup}\frac{1}{n}logr(n,ϵ)$$

and

$$h_{top-cov}\left(f_{\infty }\right)\le h_{top}\left(f_{\infty }\right).$$

Applying Lemma 5, we get

$$s(n,ϵ)\le N(\underset{i=1}{\overset{n}{\bigvee }}{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}{\mathcal{B}}_{ϵ}),$$

so

$$\underset{n\to \infty }{lim\; sup}\frac{1}{n}logs(n,ϵ)\le \underset{n\to \infty }{lim\; sup}\frac{1}{n}logN(\underset{i=1}{\overset{n}{\bigvee }}{(f_i\circ f_{i+1}\circ \dots \circ f_n)}^{-1}{\mathcal{B}}_{ϵ})$$

and finally we get the second inequality

$$h_{top}\left(f_{\infty }\right)\le h_{top-cov}\left(f_{\infty }\right).$$

The theorem is proved. □

4. Topological Entropy of L-Lipschitz Dynamical Solenoids

Dai, Zhou, and Geng [23] proved the following result. If X is a metric compact space and $f:X\to X$ a Lipschitz continuous map, then the Hausdorff dimension of X is lower estimated by the topological entropy of f divided by the logarithm of its Lipschitz constant. In 2004, Misiurewicz [24] provided a new definition of topological entropy of a single transformation, which was a kind of hybrid between the Bowen’s definition and the original definition of Adler, Konheim, and McAndrew [17]. The main theorem in [24] is similar to the result in [23]. In this section, we consider a special class of dynamical solenoids called L-Lipschitz dynamical solenoids. We say that a dynamical solenoid $f_{\infty }={(f_n:X\to X)}_{n=1}^{\infty }$ is a L-Lipschitz if there exists $L>0$ such that each map $f_n:X\to X$ is an Lipschitz epimorphism with Lipschitz constant L, i.e., for any $x,y\in X$ and arbitrary $n\in \mathbb{N}$

$$d(f_n\left(x\right),f_n\left(y\right))\le L·d(x,y).$$

Let us start with the following example.

Example 1.

Consider the dynamical solenoid $f_{\infty }={(f_n:{\mathbb{T}}^2\to {\mathbb{T}}^2)}_{n=1}^{\infty },$ where ${\mathbb{T}}^2=\frac{{\mathbb{R}}^2}{{\mathbb{Z}}^2}$ is two-dimensional torus and each $f_n:{\mathbb{T}}^2\to {\mathbb{T}}^2$ is the doubling map, i.e., $f_n(x_1,x_2)=2·(x_1,x_2),$ for any $(x_1,x_2)\in {\mathbb{T}}^2$. Then,

$$\frac{h_{top}(f_{\infty },{\mathbb{T}}^2)}{log\left(2\right)}=HD\left({\mathbb{T}}^2\right)=\frac{h_{top-dim}(f_{\infty },{\mathbb{T}}^2)}{log\left(2\right)}.$$

Indeed, the Hausdorff dimension of the two dimensional torus is equal to two (see page 23 in [25]). Due to Remarks 2 and 3, we get $h_{top}(f_{\infty },{\mathbb{T}}^2)=h_{top}\left(f_2\right)=h_{top-dim}(f_{\infty },{\mathbb{T}}^2).$ On the other hand, the doubling map $f_2:{\mathbb{T}}^2\to {\mathbb{T}}^2$ can be considered as the Cartesian product of two doubling maps $g:\frac{\mathbb{R}}{\mathbb{Z}}\to \frac{\mathbb{R}}{\mathbb{Z}}$ defined by $g\left(x\right)=2·xmod1,$ for $x\in \frac{\mathbb{R}}{\mathbb{Z}}.$ Moreover, $h_{top}\left(g\right)=log\left(2\right)$ (see Example on page 29 in [26]). Consequently, $h_{top}(f_{\infty },{\mathbb{T}}^2)=2·log\left(2\right)=h_{top-dim}(f_{\infty },{\mathbb{T}}^2).$

To show the comprehensive picture of dynamics of L-Lipschitz dynamical solenoids, we rewrite the Theorem 3 published in [11], written for nonautonomous dynamical systems, in the set up of dynamical solenoids as follows.

Theorem 3.

Assume that $f_{\infty }={(f_n:X\to X)}_{n=1}^{\infty }$ is a L-Lipschitz dynamical solenoid with $L>1$. Then, for any $Y\subset X$, we obtain

$$HD\left(Y\right)\ge \frac{h_{top-dim}(\left(f_{\infty }\right),Y)}{log\left(L\right)}.$$

For the convenience of the reader and to make the paper self-contained, we write the proof of Theorem 3 which is essentially the same as the proof of Theorem 3 in [11].

Proof. 

Choose a finite open cover $\mathcal{A}$ of Y and denote by $\delta =Leb\left(\mathcal{A}\right)$ its Lebesgue number. It means that for an open subset $C\subset Y$ with diameter $diam\left(C\right)<\delta ,$ there exists $A\in \mathcal{A}$ such that $C\subset A.$ Choose an open set B with $\frac{\delta }{L^n}\le diam\left(B\right)<\frac{\delta }{L^{n-1}},$ for some $n\in \mathbb{N}.$ We obtain that

$$diam(f_k\circ f_{k+1}\circ \dots .\circ f_{n-1}\left(B\right))<\delta$$

for any $k=1,2,\dots n-1,$ so $n_{\mathcal{A}}\left(B\right)\ge n-1.$ From the inequality

$$\frac{\delta }{L^n}\le diam\left(B\right)$$

we conclude that

$$n\ge \frac{log\left(\delta \right)-log\left(diam\right(B\left)\right)}{log\left(L\right)}.$$

Consequently,

$$\frac{log\left(\delta \right)-log\left(diam\right(B\left)\right)}{log\left(L\right)}\le n_{\mathcal{A}}\left(B\right)+1$$

and

$$dia{m}_{\mathcal{A}}\left(B\right)=exp(-n_{\mathcal{A}}\left(B\right))\le exp(1-\frac{log\left(\delta \right)-log\left(diam\right(B\left)\right)}{log\left(L\right)})=$$

$$exp[1-\left(\frac{log\left(\delta \right)}{log\left(L\right)}\right)]·{\left(diam\left(B\right)\right)}^{\frac{1}{log\left(L\right)}}.$$

Therefore, for an open cover $\mathcal{B}$ of Y consisting of open sets B with $\frac{\delta }{L^n}\le diam\left(B\right)<\frac{\delta }{L^{n-1}}$ and $\lambda >0,$ we get

$${\mathcal{D}}_{\mathcal{A}}(\mathcal{B},\lambda )\le exp[\lambda -\lambda ·\left(\frac{log\left(\delta \right)}{log\left(L\right)}\right)]·\sum _{B\in \mathcal{B}}{\left(diam\left(B\right)\right)}^{\frac{\lambda }{log\left(L\right)}}.$$

Fix $\gamma >0$ and choose ${\lambda }_1$ such that

$$\frac{{\lambda }_1}{log\left(L\right)}>HD\left(Y\right)\ge \frac{{\lambda }_1}{log\left(L\right)}-\gamma .$$

By definition of the Hausdorff measure, the equality ${\mu }_{\frac{{\lambda }_1}{log\left(L\right)}}\left(Y\right)=0$ holds. Therefore, for any $ϵ>0$ there exists and an open cover ${\mathcal{B}}_{ϵ}$ of Y such that for any $B\in {\mathcal{B}}_{ϵ}$

$$ϵ>exp[1-\left(\frac{log\left(\delta \right)}{log\left(L\right)}\right)]·{\left(diam\left(B\right)\right)}^{\frac{1}{log\left(L\right)}}>dia{m}_{\mathcal{A}}\left(B\right)$$

and

$$ϵ>exp[{\lambda }_1-{\lambda }_1·\left(\frac{log\left(\delta \right)}{log\left(L\right)}\right)]·\sum _{B\in {\mathcal{B}}_{ϵ}}{\left(diam\left(B\right)\right)}^{\frac{{\lambda }_1}{log\left(L\right)}}>D_{\mathcal{A}}({\mathcal{B}}_{ϵ},{\lambda }_1).$$

The inequalities

$${\mu }_{\mathcal{A},{\lambda }_1}\left(Y\right)\le {\mathcal{D}}_{\mathcal{A}}({\mathcal{B}}_{ϵ},{\lambda }_1)<ϵ$$

yield ${\mu }_{\mathcal{A},{\lambda }_1}\left(Y\right)=0.$ According to Definition 4, we get

$$h_{top-dim}(\left(f_{\infty }\right),Y,\mathcal{A})=inf\{\lambda :{\mu }_{\mathcal{A},\lambda }\left(Y\right)=0\}\le {\lambda }_1.$$

Taking supremum over all open finite covers of $Y,$ we obtain

$$h_{top-dim}(\left(f_{\infty }\right),Y)=$$

$$sup\{h_{top-dim}(\left(f_{\infty }\right),Y,\mathcal{A}):\mathcal{A}-finiteopencoverofY\}\le$$

$${\lambda }_1\le log\left(L\right)·(HD\left(Y\right)+\gamma ).$$

Finally,

$$h_{top-dim}(\left(f_{\infty }\right),Y)\le log\left(L\right)·HD\left(Y\right),$$

as $\gamma$ is an arbitrarily small positive number. □

In particular, taking $Y=X,$ we obtain the following corollary.

Corollary 1.

Assume that $f_{\infty }={(f_n:X\to X)}_{n=1}^{\infty }$ is a L-Lipschitz dynamical solenoid. Then, the inequality

$$HD\left(X\right)\ge \frac{h_{top-dim}\left(\left(f_{\infty }\right)\right)}{log\left(L\right)}$$

holds.

In the special case, for $f_{\infty }={(f_n:X\to X)}_{n=1}^{\infty }$ being a L-Lipschitz dynamical solenoid such that all maps $f_n:X\to X$ coincide with a continuous map $f:X\to X,$ we get that

$$h_{top}\left(f_{\infty }\right)=h_{top}\left(f\right),$$

where $h_{top}\left(f\right)$ is the classical topological entropy of $f_{:}X\to X.$ Bowen proved (Proposition 1 in [20]) that $h_{top-dim}\left(f\right)=h_{top}\left(f\right).$ Consequently, as a corollary of Theorem 3, we get the result of Misiurewicz [24].

Corollary 2

(Theorem 2.1 in [24]). If $f:X\to X$ is a continuous L-Lipschitz map of a compact metric space $(X,d)$, then

$$HD\left(X\right)\ge \frac{h_{top}\left(f\right)}{log\left(L\right)}.$$

5. Topological Entropy of Locally Expanding Dynamical Solenoids

In this section, we investigate locally expanding dynamical solenoids. Ruelle [16] introduced the notion of a locally expanding map in the following way.

Definition 5.

Let $(X,d)$ be a compact metric space and $f:X\to X$ a continuous selfmap. If for $\lambda >1$ there exists $ϵ>0$ such that for every pair of distinct points $x,y\in X$

$$d(x,y)<ϵ\Rightarrow d\left(f\right(x),f(y\left)\right)\ge \lambda ·d(x,y),$$

then we say that f is a locally $(ϵ,\lambda )$-expanding map and λ is an expanding coefficient of f.

Notice that any finite composition of locally $({ϵ}_i,{\lambda }_i)$-expanding maps is an $(ϵ,\lambda )$-locally expanding map for some $ϵ>0$ and $\lambda >1.$ We extend the notion of locally expanding map to a dynamical solenoid as follows.

Definition 6.

Given a dynamical solenoid $f_{\infty }={(f_n:X\to X)}_{n=1}^{\infty }$. If there exists $ϵ>0$ such that all maps $f_n:X\to X$ are locally $(ϵ,{\lambda }_n)$-expanding and $\lambda :=inf\{{\lambda }_n:n\in \mathbb{N}\}>1,$ then we say that $f_{\infty }$ is locally λ-expanding.

Lemma 6.

Given a locally λ-expanding dynamical solenoid $f_{\infty }={(f_n:X\to X)}_{n=1}^{\infty }$. Then, there exists $ϵ>0$ such that for any $x\in X,k\in \mathbb{N},$ and $\gamma \in (0,ϵ)$ we get

$$\bigcap _{i=1}^k{(f_i\circ f_{i+1}\circ \dots \circ f_k)}^{-1}B(f_i\circ f_{i+1}\circ \dots \circ f_k\left(x\right),\gamma )\subset B\left(x,\frac{\gamma }{{\lambda }^k}\right).$$

Proof. 

Choose $ϵ>0$ such that for any $k\in \mathbb{N}$ and for every pair of distinct points $x,y\in X,$ we get

$$d(x,y)<ϵ\Rightarrow d(f_k\left(x\right),f_k\left(y\right))\ge {\lambda }_k·d(x,y).$$

Fix $\gamma \in (0,ϵ)$ and let

$$y\in \bigcap _{i=1}^k{(f_i\circ f_{i+1}\circ \dots \circ f_k)}^{-1}B(f_i\circ f_{i+1}\circ \dots \circ f_k\left(x\right),\gamma ).$$

Then, we get inequalities

$$ϵ>\gamma >d(f_1\circ f_2\circ \dots \circ f_k\left(x\right),f_1\circ f_2\circ \dots \circ f_k\left(y\right))\ge$$

$${\lambda }_1·d(f_2\circ \dots \circ f_k\left(x\right),f_2\circ \dots \circ f_k\left(y\right))\ge {\lambda }_1·\dots ·{\lambda }_k·d(x,y)\ge$$

$${\lambda }^k·d(x,y).$$

Therefore, $d(x,y)<\frac{\gamma }{{\lambda }^k}$ and $y\in B(x,\frac{\gamma }{{\lambda }^k}).$ The lemma is proved. □

The notion of the box dimension is an example of fractal dimension which belongs to fractal geometry. It was defined independently by Minkowski and Bouligard for a subset of Euclidean space. For modern presentation of fractal dimensions see the classical books of Falconer [25,27] or the monograph written by Przytycki and Urbański [28].

Definition 7

(Chapter 2 in [25]). Recall that the upper box dimension of a closed subset Z of a compact metric space X is

$$\overline{{dim}_B\left(Z\right)}:=\underset{\gamma \to 0}{lim\; sup}\frac{logN(Z,\gamma )}{-log\gamma },$$

where $N(Z,\gamma )$ denotes the smallest number of balls $B(x,\gamma )$ of radius $\gamma >0$ needed to cover Z.

Lemma 7

([28]). For a compact metric space $X,$ the Hausdorff dimension $HD\left(X\right)$ of X and the upper box dimension $\overline{{dim}_B\left(X\right)}$ of X are interrelated

$$HD\left(X\right)\le \overline{{dim}_B\left(X\right)}.$$

In the proof of Theorem 4 we need the following lemma.

Lemma 8

(Lemma 6.2 in [29]). Let $\varphi :R\to R_{+}$ be a decreasing function. If $\delta \in (0,1)$ and $\gamma >0$, then

$$\underset{r\to 0}{lim\; sup}\frac{log\varphi \left(r\right)}{logr}=\underset{n\to \infty }{lim\; sup}\frac{log\varphi \left({\delta }^n\gamma \right)}{log\left({\delta }^n\gamma \right)}.$$

Theorem 4.

Given a locally λ-expanding dynamical solenoid $f_{\infty }={(f_n:X\to X)}_{n=1}^{\infty }$. Then,

$$h_{top}\left(f_{\infty }\right)\ge (log\lambda )·\overline{{dim}_B\left(X\right)}\ge (log\lambda )·HD\left(X\right).$$

Proof. 

In the first part of the proof we intend to show that

$$h_{top}\left(f_{\infty }\right)\ge (log\lambda )·\overline{{dim}_B\left(X\right)}.$$

(1)

Fix $ϵ>0$ such that for every pair of distinct points $x,y\in X$ and for every $n\in \mathbb{N}$,

$$d(x,y)<ϵ\Rightarrow d(f_n\left(x\right),f_n\left(y\right))\ge \lambda ·d(x,y).$$

By Lemma 6 and Lemma 1, for any $\gamma \in (0,ϵ)$ and an arbitrary $n\in \mathbb{N}$, we have

$$N\left(X,\frac{\gamma }{{\lambda }^n}\right)\le r(n,\gamma ),$$

(2)

consequently, applying Lemma 8 for the first equality and (2) for the subsequent inequality, we get

$$\overline{{dim}_B\left(X\right)}=\underset{n\to \infty }{lim\; sup}\frac{logN(X,\frac{\gamma }{{\lambda }^n})}{-log\frac{\gamma }{{\lambda }^n}}\le \underset{n\to \infty }{lim\; sup}\frac{logr(n,\gamma )}{-log\frac{r}{{\lambda }^n}}=$$

$$\frac{1}{log\lambda }·\underset{n\to \infty }{lim\; sup}\frac{logr(n,\gamma )}{n}.$$

Therefore,

$$h_{top}\left(f_{\infty }\right)=\underset{\gamma \to 0}{lim}\underset{n\to \infty }{lim\; sup}\frac{logr(n,\gamma )}{n}\ge (log\lambda )·\overline{{dim}_B\left(X\right)}.$$

According to the Lemma 7, we finally get

$$h_{top}\left(f_{\infty }\right)\ge (log\lambda )·\overline{{dim}_B\left(X\right)}\ge (log\lambda )·HD\left(X\right).$$

 □