Hausdorff Dimension and Topological Entropies of a Solenoid

The purpose of this paper is to elucidate the interrelations between three essentially different concepts: solenoids, topological entropy, and Hausdorff dimension. For this purpose, we describe the dynamics of a solenoid by topological entropy-like quantities and investigate the relations between them. For L-Lipschitz solenoids and locally λ—expanding solenoids, we show that the topological entropy and fractal dimensions are closely related. For a locally λ—expanding solenoid, we prove that its topological entropy is lower estimated by the Hausdorff dimension of X multiplied by the logarithm of λ.


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, we investigate the complexity of a solenoid determined by the sequence f ∞ = ( f n : X → X) ∞ n=1 of continuous epimorphisms of a compact metric space (X, d), called bonding maps. By solenoid determined by f ∞ , we mean the inverse limit As X ∞ is uniquely determined by f ∞ , we will use these two terms interchangeable. A solenoid is both a compact connected metric space (continuum) and a dynamical object of complicated structure. If additionally X is an abelian group then the compact metric space X ∞ is an abelian group as well. 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].
Solenoids are a natural generalization of classical dynamical systems. 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 solenoid f ∞ : topological entropy h top ( f ∞ ), topological cover entropy h top−cov ( f ∞ ), and topological dimensional entropy h top−dim ( f ∞ ).
Both nonautonomous dynamical systems and solenoids are determined by compositions of continuous self-maps; therefore, in both cases, the entropy-like quantities that capture complexities of dynamical systems can be similar. For example, the topological entropy of a 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 solenoid which were previously known for continuous maps on compact metric spaces, and we obtained the following results.
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 > 0, is it possible to compute the entropy with maximal error of ?" 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 solenoids, we rewrite the Theorem 3 in [11] to express complexity of so called L-Lipschitz solenoid. A solenoid f ∞ = ( f n : X → X) ∞ n=1 is called L-Lipschitz if it consists of L-Lipschitz epimorphisms; the following inequality holds.
where HD(Y) is the Hausdorff dimension of Y.
Finally, we investigate so called locally λ−expanding solenoids, in the sense of Ruelle [16] (see Definition 5). We prove that the topological entropy of such a solenoid, defined on the space X, is related to the upper box dimension of X multiplied by the logarithm of λ. We obtained the following inequalities.

Theorem 4. Given a locally λ−expanding
The paper is organized as follows. In Section 2, we introduce several definitions of entropy-like quantities for a 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 solenoids; we present Theorem 3. Finally, in Section 5, we investigate locally λ−expanding solenoids and prove Theorem 4.

Topological Entropies of a 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 → 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 solenoids.
In the paper, we consider a solenoid determined by a sequence f ∞ = ( f n : X → X) ∞ n=1 of continuous epimorphisms of a compact metric spaces (X, d). Thus, we obtain that the solenoid is a generalized dynamical system. Its complexity, complicated topological structure, and chaos can be measured by several entropy-like quantities. First, we introduce topological entropy via (n, )−separated sets. Let B(x, r) = {y ∈ X : d(x, y) ≤ r} denote a closed ball in the metric space (X, d) centered at x ∈ X and with radius r.
A set E ⊂ X is called (n, )-separated if for any pair of distinct points x, y ∈ E we have The following two lemmas are a reformulation of Definition 1.
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 solenoids as follows.

Definition 2. The quantity
is called the topological entropy of the solenoid f ∞ .

Remark 1.
The topological entropy of a solenoid can also be expressed in the language of (n, )-spannings sets.
Consequently, passing to the suitable limits, we obtain the equality h top ( f ∞ ) = lim →0 + lim sup n→∞ 1 n log r(n, ).

Remark 2.
Assume that all maps of the sequence f ∞ = ( f n : X → X) ∞ n=1 coincide with a fixed continuous map f : X → X of a compact metric space (X, d). Then, the topological entropy of f ∞ is equal to the topological entropy of f . For example, the topological entropy of a solenoid coincides with the topological entropy of a nonautonomous dynamical system defined in [11].

Topological Entropy of a Solenoid via Open Covers
It is a well-known fact that topological entropy of a single continuous map f : X → 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 solenoids. For this purpose, notice that for two open covers A, B of X, the family is an open cover of X. Moreover, for a continuous map is an open cover of X. Thus, for the open cover A of X, the family is an open cover of X.
For an open cover B of X let us denote by N(B) the number of sets in a finite subcover of B covering X, with the smallest cardinality.
whereas the topological cover entropy of f ∞ is the quantity where A ranges over all open covers of X.

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 ∞ . For any λ > 0 the classical Hausdorff λ−measure µ λ (Y) of a subset Y ⊂ X is defined as follows, The function λ → µ λ (Y) has a unique critical point, where it jumps from ∞ to 0. The Hausdorff dimension HD(Y) of Y is defined as the critical point of the function λ → µ λ (Y), i.e.,

Generalized Hausdorff Measure and Generalized Hausdoff Dimension
Arguments similar to the construction of the classical Hausdorff λ-measure and the Hausdorff dimension lead to another entropy-like quantity for The behavior of the function λ → µ A,λ (Y) is very similar to the behavior of λ → µ λ (Y): it has a unique critical point, where it jumps from ∞ to 0. More precisely.
The number Remark 3. Our definition of topological dimension entropy of a solenoid is an extension of Bowen's entropy [20]. Moreover, the topological dimensional entropy of a solenoid is just Bowen's topological entropy of nonautonomous dynamical systems in [22].

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

Proof. Choose a finite open cover A of X and let
Denote by B n a finite subcover of A n with cardinality |B n | = N(A n ). Then, for any B ∈ B n , we obtain that n A (B) ≥ n, so diam A (B) ≤ exp(−n) and for any λ > 0 we get As |B n | = N(A n ), we have |B n | · exp(−λ · n) = exp(−λ · n + log |B n |) = exp(−n(λ − 1 n log N(A n ))). Consequently, Fix > 0 and arbitrary small γ > 0. Choose λ * such that For sufficiently large n ∈ N, we obtain the inequalities As > 0 is arbitrarily small, the above two inequalities yield µ A,λ * (X) = 0. Therefore, As A is an arbitrary finite open cover of X, we obtain Finally, passing with γ to zero, we get .
Proof. Fix n ∈ N. Choose an (n, δ 2 )-spanning set F with cardinality card(F) = r(n, δ 2 ). As Leb(A) = δ, we obtain that any ball B is a subset of some member of the covering On the other hand, applying Lemma 1, we get .
Proof. Choose an (n, )-separated set E with cardinality card(E) = s(n, ). Assume that two distinct points x 1 , x 2 ∈ E belong to the same member of the cover n i=1 On the other hand, as the set E is (n, )-separated, there exists j ∈ {1, ..., n} such that Thus, we get a contradiction with diam(B j ) ≤ . Therefore, Now, we are ready to prove that the topological entropy of a solenoid is equivalent to its topological covering entropy.
Proof. Fix > 0. Let A be the cover of X by all open balls of radius 2 · and denote by B the cover of X by all open balls of radius 2 . Due to Lemma 4, we obtain and Applying Lemma 5, we get and finally we get the second inequality The theorem is proved.

Topological Entropy of L-Lipschitz Solenoids
Dai, Zhou, and Geng [23] proved the following result. If X is a metric compact space and f : X → 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 solenoids called L-Lipschitz solenoids. We say that a solenoid determined by f ∞ = ( f n : X → X) ∞ n=1 is a L-Lipschitz if there exists L > 0 such that each map f n : X → X is an Lipschitz epimorphism with Lipschitz constant L, i.e., for any x, y ∈ X and arbitrary n ∈ N d( f n (x), f n (y)) ≤ L · d(x, y).
Let us start with the following example.
is two-dimensional torus and each f n : T 2 → 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 ) ∈ T 2 . Then, 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 ∞ , . On the other hand, the doubling map f 2 : T 2 → T 2 can be considered as the Cartesian product of two doubling maps g : R Z → R Z defined by g(x) = 2 · x mod 1, for x ∈ R Z . Moreover, h top (g) = log(2) (see Example on page 29 in [26]). Consequently, To show the comprehensive picture of dynamics of L-Lipschitz solenoids, we rewrite the Theorem 3 published in [11], written for nonautonomous dynamical systems, in the set up of solenoids as follows.
Theorem 3. Assume that f ∞ = ( f n : X → X) ∞ n=1 is a L-Lipschitz solenoid with L > 1. Then, for any Y ⊂ X, we obtain 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 A of Y and denote by δ = Leb(A) its Lebesgue number. It means that for an open subset Choose an open set B with δ L n < diam(B) < δ L n−1 . We obtain that we conclude that Fix γ > 0 and choose λ 1 such that The inequalities According to Definition 4, we get Taking supremum over all open finite covers of Y, we obtain Finally, as γ is an arbitrarily small positive number.
In particular, taking Y = X, we obtain the following corollary.
In the special case, for f ∞ = ( f n : X → X) ∞ n=1 being a L-Lipschitz solenoid such that all maps f n : X → X coincide with a continuous map f : X → X, we get that where h top ( f 2 ) is the classical topological entropy of f 2 : X → X. Bowen proved (Proposition 1 in [20]) that h top−dim ( f 2 ) = h top ( f 2 ). Consequently, as a corollary of Theorem 3, we get the result of Misiurewicz [24]. Corollary 2 (Theorem 2.1 in [24]). If f : X → X is a continuous L-Lipschitz of a compact metric space (X, d), then

Topological Entropy of Locally Expanding Solenoids
In this section, we investigate the dynamics of locally expanding 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 → X a continuous selfmap. If for λ > 1 there exists > 0 such that for every pair of distinct points x, y ∈ X d(x, y) < ⇒ d( f (x), f (y)) ≥ λ · d(x, y), then we say that f is a locally ( , λ)-expanding map and λ is an expanding coefficient of f .
Notice that any finite composition of locally ( i , λ)-expanding maps is an ( , λ)-locally expanding map for some > 0. We extend the notion of locally expanding map to a solenoid as follows.

Definition 6.
Given a solenoid f ∞ = ( f n : X → X) ∞ n=1 , if all maps f n : X → X are locally ( n , λ)-expanding, then we say that the solenoid f ∞ is locally λ-expanding. Lemma 6. Given a locally λ-expanding solenoid determined by a sequence f ∞ = ( f n : X → X) ∞ n=1 . Then, for any k ∈ N, there exists δ k > 0 such that for every pair of distinct points x, y ∈ X Moreover, for any γ ∈ (0, δ k ),

Proof.
A composition of k locally expanding maps is again a locally expanding map. Therefore, there exists δ k > 0 such that for every pair of distinct points x, y ∈ X, we get for every i ∈ {1, 2, ..., k}. If y = x, then clearly x ∈ B x, γ λ k . Therefore, assume that y = x, as d(x, y) < γ < δ k , we get inequalities Therefore, d(x, y) < γ λ k and y ∈ B(x, γ λ 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]. In the proof of Theorem 4 we need the following lemma.
Proof. In the first part of the proof we intend to show that Let > 0 and λ > 1 be such that for every pair of distinct points x, y ∈ X and for every i ∈ {1, . . . , k}, d(x, y) < ⇒ d( f i (x), f i (y)) ≥ λ · d(x, y).