Topological entropy of free semigroup actions for noncompact sets

In this paper we introduce the topological entropy and lower and upper capacity topological entropies of a free semigroup action, which extends the notion of the topological entropy of a free semigroup action defined by Bufetov [ 10 ], by using the Caratheodory-Pesin structure (C-P structure). We provide some properties of these notions and give three main results. The first is the relationship between the upper capacity topological entropy of a skew-product transformation and the upper capacity topological entropy of a free semigroup action with respect to arbitrary subset. The second are a lower and a upper estimations of the topological entropy of a free semigroup action by local entropies. The third is that for any free semigroup action with $m$ generators of Lipschitz maps, topological entropy for any subset is upper bounded by the Hausdorff dimension of the subset multiplied by the maximum logarithm of the Lipschitz constants. The results of this paper generalize results of Bufetov [ 10 ], Ma et al. [ 26 ], and Misiurewicz [ 27 ].


Introduction. Topological entropy was first introduced by Adler, Konheim and
McAndrew [1]. Later, Bowen [7] and Dinaburg [17] defined topological entropy for a uniformly continuous map on metric space and proved that for a compact metric space, they coincide with that defined by Adler et al. Since the topological entropy appeared to be a very useful invariant in ergodic theory and dynamical systems, there were several attempts to find its suitable generalizations for other systems such as groups, pseudogroups, graphs, foliations, nonautonomous dynamical systems and so on [3,4,5,6,10,11,12,13,19,20,21,22,23,25,34,35]. Bowen [8] extended the concept of topological entropy for non-compact sets in a way which resembles the Hausdorff dimension. Pesin [28] gave a new characterization of topological entropy and topological pressure for non-compact sets by Carathéodory structure, which we call Carathéodory-Pesin structure or C-P structure for short. Topological entropy for non-compact sets can be used to investigate multifractal spectra and saturated It was shown in [28] that there exists a critical value α C ∈ [−∞, ∞] such that m(Z, α) = 0, α > α C , m(Z, α) = ∞, α < α C .
The number α C is called the Carathéodory-Pesin dimension of the set Z. Now we assume that the following condition holds: (3 ) there exists > 0 such that for any 0 < ε ≤ there exists a finite or countable subcollection G ⊂ S covering X such that ψ(s) = ε for any s ∈ G.
The numbers α C and α C are called the lower and upper Carathéodory-Pesin capacities of the set Z respectively.
For any ε > 0 and subset Z ⊂ X, put where the infimum is taken over all finite or countable subcollections G ⊂ S covering Z such that ψ(s) = ε for any s ∈ G.
Example 1. Let (X, d) be a metric space, Z ⊂ X. Setting F the collection of the balls {B(x, ε) : x ∈ X, ε > 0}, S = {(x, ε) : x ∈ X, ε > 0}, ξ(x, ε) ≡ 1, and η(x, ε) = ψ(x, ε) = diamB(x, ε), we can get where the infimum is taken over all finite or countable subcollections G ⊂ S covering Z with ε i < ε. Then α c equals to the Hausdorff dimension of Z. Moreover, the lower and upper Carathéodory-Pesin capacities of Z also equal to the lower and upper box dimensions of the set Z respectively.

Words and sequences.
Let F + m be the set of all finite words of symbols 0, 1, . . . , m − 1. For any w ∈ F + m , |w| stands for the length of w, that is, the digits of symbols in w. Obviously, F + m with respect to the law of composition is a free semigroup with m generators. We write w ≤ w if there exists a word w ∈ F + m such that w = w w . For w = i 1 . . . i k ∈ F + m , denote w = i k . . . i 1 . Let Σ m be the set of all two-side infinite sequences of symbols 0, 1, . . . , m − 1, that is The metric on Σ m is defined by Obviously, Σ m is compact with respect to this metric. The bernoulli shift σ m : Σ m → Σ m is a homeomorphism of Σ m given by the formula Let Σ + m be the set of all one-side infinite sequences of symbols 0, 1, . . . , m − 1: Topological entropy for a free semigroup action in [10]. In this section, we recall the topological entropy for a free semigroup action. Our presentation follows Bufetov [10].
Let X be a compact metric space with metric d. Suppose a free semigroup with m generators acts on X. Denote the maps corresponding to the generators by f 0 , f 1 , . . . , f m−1 . Assume that these maps are continuous. Let Let ε > 0, a subset E of X is said to be a (w, ε, f 0 , . . . , f m−1 )-spanning subset if for any x ∈ X, there exists y ∈ E with d w (x, y) < ε. The minimal cardinality of a (w, ε, f 0 , . . . , f m−1 )-spanning subset of X is denoted by B(w, ε, f 0 , . . . , f m−1 ).
A subset F of X is said to be a (w, ε, f 0 , . . . , f m−1 )-separated subset if for any x, y ∈ F, x = y implies d w (x, y) ≥ ε. The maximal cardinality of a (w, ε, f 0 , . . . , In [10], Bufetov defined the topological entropy of a free semigroup action by the formula h(f 0 , . . . , f m−1 ) = lim 2.4. Local entropy. Let (X, d) be a compact metric space, f : X → X a continuous map and µ a Borel probability measure on X. For any n ≥ 1, ε > 0 and x ∈ X, define the Bowen ball centered at x by Brin and Katok [9] introduced the notion of local entropy for a single map f in the following way.
is called the lower local entropy of µ at point x ∈ X while the quantity is called the upper local entropy of µ at point x ∈ X.
2.5. Hausdorff dimension. Let (X, d) be a metric space. Given a subset Z ⊂ X, s ≥ 0 and δ > 0, we define (diam(U )) s : B is a cover of Z and diam(U ) < δ for all U ∈ B .
As δ decreases, H s δ (Z) increases, therefore there exists the limit H s (Z) = lim 3. Topological entropy, lower and upper capacity topological entropies of a free semigroup and their properties. In this section, we introduce the definitions of topological entropy, lower and upper capacity topological entropies of a free semigroup by using C-P structure and provide some properties of them.
Given w ∈ F + m , |w| = N, Z ⊂ X and α ≥ 0, we define where the infimum is taken over all finite or countable collections of strings G w ⊂ S(U) such that m(U) ≥ N + 1 for all U ∈ G w and G w covers Z (i.e., for any We can easily verify that the function M (Z, α, U, N ) is non-decreasing as N increases. Therefore there exists the limit Furthermore, given w ∈ F + m and |w| = N , by the Condition (3 ) in section 2.1, we can define Gw {card(G w )}, the infimum is taken over all finite or countable collections of strings G w ⊂ S(U) such that m(U) = N + 1 for all U ∈ G w and G w covers Z (i.e., for any U ∈ G w , there is w U ∈ F + m such that w U = w and It is easy to see that R(Z, α, U, N ) = Λ(Z, U, N ) exp(−α(N + 1)).
We set The C-P structure τ generates the Carathéodory-Pesin dimension of Z and the lower and upper Carathéodory-Pesin capacities of Z with respect to G. We denote them by h Z (U, G), Ch Z (U, G), and Ch Z (U, G) respectively. We have Theorem 3.1. For any set Z ⊂ X, the following limits exist: Proof. We use the analogous method as that of [28] . Let V be a finite open cover of X with diameter smaller than the Lebesgue number of U. Then each element Then for every α > 0 and N > 0. One can easily see that . Since X is compact it has finite open covers of arbitrarily small diameter. Therefore, This implies the existence of the first limit. The existence of the two other limits can be proved in similar ways.
The quantities h Z (G), Ch Z (G), and Ch Z (G) are called the topological entropy and lower and upper capacity topological entropies of G on the set Z respectively.
(2) Indeed, let f : X → X be a continuous transformation and Ch Z (f ) are the topological entropy and lower and upper capacity topological entropies defined by Pesin [28].
, the classical topological entropy defined by Adler et al [1].

3.2.
Properties of topological entropy and lower and upper capacity topological entropies. Using the basic properties of the Carathéodory-Pesin dimension [28] and definitions, we get the following basic properties of topological entropy and lower and upper capacity topological entropies of a free semigroup action.
Obviously, the function η and ψ satisfy Condition (4) in section 2.1. Therefore, similar to the Theorem 2.2 in [28], we obtain the following lemma.
Proof. We will prove the first equality; the second one can be proved in a similar fashion. Put It follows that R(Z, α + γ, U, N i ) ≤ 1 for all sufficiently large i. Therefore, for such numbers i, Therefore, Let us now choose a sequence N i such that Taking the limit as i → ∞ we obtain that Since γ can be chosen arbitrarily small the inequalities (1) and (2) imply that α = β.
Remark 3. By the Theorem 3.1 and Lemma 3.2, we can obtain For a free semigroup with m generators acting on X, denote the maps corresponding to the generators by For invariant sets, similar to the lower and upper capacity topological entropies of a single map [28], we have the following theorems.
moreover, for any open cover U of X, we have  (2) . Then and m(UV) = m(U) + m(V). Since Z is a G-invariant set, the collection of strings G w also covers Z. By the definition of Λ w (Z, U, p + q + 1). We have  Since M (Z, α, U, N ) is non-decreasing as N increases and non-negative, it follows that M (Z, α, U, N ) = 0 for any N . Therefore, for any w ∈ F + m and |w| = N , we have M w (Z, α, U, N ) = 0. For M w (Z, α, U, 2) = 0, there exists A w ⊂ S(U) such that A w covers Z (i.e., for any U ∈ A w , there exists w U ∈ F + m such that where p is a constant. Since Z is compact we can choose A w to be finite and K ≥ 3 to be a constant and For any w (1) , w (2) ∈ F + m , |w (1) | = |w (2) | = 2 and j ∈ {0, 1, . . . , m − 1}, we can construct (3), (4). Then where the word corresponds to UV is w U jw V and m(UV) = m(U) + m(V) ≥ 6. Since Z is G-invariant, then A w (1) jw (2) covers Z. It is easy to see that By mathematical induction, for each n ∈ N and j 1 , . . . , j n−1 ∈ {0, 1, . . . , m − 1}, we can define A w (1) j1w (2) j2...w (n−1) jn−1w (n) which covers Z and satisfies (2) ... covers Z and Therefore, for any ω ∈ Σ + m , there exists Γ ω covering Z and Q(Z, α, Γ ω ) < ∞. Put F = {Γ ω : ω ∈ Σ + m }. Condition 3.5. For any N > 0 and any w = i 1 i 2 . . . i N ∈ F + m , there exists Γ ω ∈ F such that for any U ∈ Γ ω , w ≤ w U and N + 1 ≤ m(U) ≤ N + K, where w U is the word corresponds to U and K is a constant as that in (4). Proof. Under the condition 3.5. For any N > 0 and any w = i 1 i 2 . . . i N ∈ F + m , there is Γ ω ∈ F covering Z such that for any U ∈ Γ ω , the word corresponds to U is w U and w ≤ w U . Then for any x ∈ Z, there exists a string U = (U 0 , U 1 , . . . , U N , . . . , U N +P ) ∈ Γ ω such that x ∈ X w U (U), where 0 ≤ P < K. Let U * = {U 0 , U 1 , . . . , U N }. Then X w U (U) ⊂ X w (U * ). If Γ * w denotes the collection of all substrings U * constructed above then Therefore, By Lemma 3.2 we obtain that α > Ch Z (U, G), and hence the desired result follows Remark 2(1).

4.
Two equivalent definitions of topological entropy in the present paper. Now, we describe two other approaches to redefine the topological entropy and lower and upper capacity topological entropies of G = {f 0 , . . . , f m−1 } on any subset of X.
Theorem 4.1. For any set Z ⊂ X, the following limits exist: Proof. Let U be a finite open cover of X, and δ(U) is the Lebesgue number of U. It is easy to see that for every x ∈ X, if x ∈ X ω (U) for some U ∈ S k+1 (U) and some ω ∈ Σ + m then Hence Let |U| → 0, then δ(U) → 0 and hence the theorem is proved. The existence of the two other limits can be proved in a similar fashion.

4.2.
Definition using Bowen's approach. Let (X, d) be a compact metric space and f : X → X be a continuous map. Bowen [8] used a way which resembles the Hausdorff dimension to construct the topological entropy of f for non-compact sets.
In the following, we give the topological entropy of free semigroup actions for noncompact sets by Bowen's approach [8] and prove that the new definition is equivalent to the definition of topological entropy in section 3. Given Z ⊂ X, ε ∈ (0, 1) and α ≥ 0, let N = [− log(ε)] + 1, where [− log(ε)] is the integer portion of − log(ε) and w = i 1 i 2 . . . i N ∈ F + m , we define where the infimum is taken over all covers B of Z such that for any B ∈ B there exists We can easily verify that the function µ(Z, α, U, ε) is non-decreasing as ε increases. Therefore, there exists the limit µ(Z, α, U) = lim  In order to prove this theorem, we propose the following lemma.
Proof. According to the definition of M w (Z, α, U, N ), we choose collection of strings G w ⊂ S(U) such that m(U) ≥ N + 1 for all U ∈ G w and G w covers Z. For any U = (U i0 , U i1 , . . . , U i m(U)−1 ) ∈ G w , then there exists ω U ∈ Σ + m such that ω U | [0,N −1] = w and we associate the set Hence On the other hand, for any λ > µ w (Z, α, U, ε), there is a cover B of Z such that In the following, we discuss two cases of n ω B ,U (B). . . .
Then G B is a collection of strings that covers Z and for any U ∈ G B , we have m(U) ≥ N + 1.
Moreover, we obtain that M w (Z, α, U, N ) < 2λ, and consequently, The proof of Theorem 4.2. According to Lemma 4.3, it follows that By the definitions of h Z (G) and h Z (G), we have Letting |U| → 0, and the theorem is proved. For any subset Z ⊂ X, w ∈ F + m and ε > 0, a subset E ⊂ X is said to be a (w, ε, Z, f 0 , . . . , f m−1 )-spanning set of Z, if for any x ∈ Z, there exists y ∈ E such that d w (x, y) < ε. Define B(w, ε, Z, f 0 , . . . , f m−1 ) to be the minimum cardinality of any (w, ε, Z, f 0 , . . . , f m−1 )-spanning sets of Z. A subset F ⊂ Z is said to be a (w, ε, Z, f 0 , . . . , f m−1 )-separated set of Z, if x, y ∈ F , x = y implies d w (x, y) ≥ ε. Let N (w, ε, Z, f 0 , . . . , f m−1 ) denotes the maximum cardinality of any (w, ε, Z, f 0 , . . . , f m−1 )-separated sets of Z. Remark 5. If Z = X , then Ch X (G) = Ch X (G) = h(G). Thus upper and lower capacity topological entropies of a free semigroup action on X are as same as the topological entropy of the free semigroup action on X defined by Bufetov [10].
A skew-product transformation F : Σ m × X → Σ m × X is defined by the formula where ω = (. . . , ω −1 , ω 0 , ω 1 , . . .) ∈ Σ m . Here f ω0 stands for f 0 if ω 0 = 0, and for f 1 if ω 0 = 1, and so on. Let ω = (. . . , ω −1 , ω 0 , ω 1 , . . .) ∈ Σ m , and the metric D on Σ m × X is defined as For F : Σ m × X → Σ m × X. Let G = {F }, we can get a C-P structure on Σ m × X. For any set Z ⊂ X, from Pesin [28], we call Ch Σm×Z (F ) the upper capacity topological entropy of F on the set Σ m × Z. Our purpose is to find the relationship between the upper capacity topological entropy Ch Σm×Z (F ) of the skew-product transformation F and the upper capacity topological entropy Ch Z (G) of a free semigroup action generated by G = {f 0 , . . . , f m−1 }. To prove this theorem, we first give the following two lemmas. The proofs of these two lemmas are simliar to that of Bufetov [10]. Therefore, we omit the proof. In this section, we give the concepts of lower and upper local entropies. Moreover, similar as the proof in Ma and Wen [26], we get two estimates. Definition 6.1. Let µ be a Borel probability measure on X. Then we call the L + lower local entropy of µ at point x with respect to G, while the quantity is called L − lower local entropy of µ at point x with respect to G.
For any N ≥ N * , take any w ∈ F + m , |w| = N . Set a cover of Z * : For each i, there exists an x i ∈ Z * ∩ B ωi| [0,n i −1] (y i , r 2 ). By the triangle inequality In combination with (7), we can get Therefore, M w (Z * , s − ε, r, N ) ≥ µ(Z * ) > 0 for all N ≥ N * , and we obtain and consequently which in turn implies that h Z * (r, G) ≥ s − ε. Then we have h Z * (G) ≥ s − ε by letting r → 0. It follows that h Z (G) ≥ h Z * (G) ≥ s − ε and hence h Z (G) ≥ s since ε > 0 is arbitrary. The proof is completed now. Proof. The proof follows [26] and is omitted. Theorem 6.4. Let µ denote a Borel probability measure on X, Z be a Borel subset of X and s ∈ (0, ∞). If Proof. Since h L + µ,G (x) ≤ s for all x ∈ Z, then for any ω ∈ Σ + m and x ∈ Z, For any N ≥ 1 and for all r ∈ (0, 1/k), ω ∈ Σ + m and ω| [0,N −1] = w . Now fix k ≥ 1 and 0 < r < 1 3k . For each x ∈ Z k , we take ω x ∈ Σ + m such that ω x | [0,N −1] = w, there exists a strictly increasing sequence {n j (x)} ∞ j=1 such that for all j ≥ 1. So, the set Z k is contained in the union of the sets in the family By Lemma 6.3, there exists a subfamily G w = {B ωx i | [0,n i −1] (x i , r)} i∈I ⊂ F w consisting of disjoint balls such that for all i ∈ I.
The index set I is at most countable since µ is a probability measure and G w is a disjointed family of sets, each of which has positive µ-measure. Therefore, which in turn implies that h Z k (3r, G) ≤ s + ε for any 0 < r < 1 3k . Letting r → 0 yields h Z k (G) ≤ s + ε for any k ≥ 1.
Remark 9. For the topological entropy of a pseudogroup introduced in [5], Biś proved a similar result to the Theorem 6.2. In [12], the authors obtained a similar result using the skew product.
Remark 10. If m = 1, then the above theorems coincide with the main results that Ma and Wen proved in [26] . 7. Topological entropy and hausdorff dimension. In dynamical systems, the relation between topological entropy and Hausdorff dimension is a very important problem which many people have studied, such as Dai et al. [16], Misiurewicz [27], Ma and Wu [24], etc. Let X be a nonempty compact metric space. Supposing that a free semigroup with m generators G = {f 0 , f 1 , . . . , f m−1 } acts on X, we assume that these maps are Lipschitz self-maps with Lipschitz constant L i respectively. In this section, we show that the topological entropy of a free semigroup action generated by G on any subset Z of X is upper bounded by Hausdorff dimension of Z multiplied by the maximum logarithm of {L i }.
The number c = log δ log L is a constant, as long as G and U are fixed. We can rewrite (8) as D ω,U (B) = exp(−n ω,U (B)) ≤ e −c (diam(B)) 1/ log L , where ω ∈ Σ + m . Therefore if B is a cover of Z, then for any λ > 0, w = i 1 i 2 . . . i N ∈ F + m and every B ∈ B with ω B ∈ Σ + m such that ω B | [0,N −1] = w, we have (1) Let f : X → X be a Lipschitz continuous map with Lipschitz constant L > 1. If G = {f }, for any Z ⊂ X, Misiurewicz [27] proved that h Z (G) ≤ HD(Z) · log L.
(2) In the future we will introduce the topological pressure of free semigroup actions for noncompact sets.