and Jumi Oh * Topological entropy for positively weak measure expansive shadowable maps

The main goal of the study on dynamical systems is to understand the structure of the orbits for homeomorphisms or ows on a compact metric space. To describe the dynamics on the underlying space, it is common to study the dynamic properties such as shadowing property, expansiveness, entropy, etc. It has close relations with stable or chaotic and sensitive properties of a given system. Recently, Morales [1] has introduced the notion of measure expansiveness, generalizing the concept of expansiveness, and Lee et al. [2] has introduced a notion of weak measure expansiveness for ows which is really weaker than measure expansive ows in [3]. The concept of positively measure-expansiveness is introduced by [1] as a generalization of the notion of positively expansiveness, and positively measure expansive continuousmaps of a compactmetric space are studied from themeasure theoretical point of view. Also Morales [4] proved that every homeomorphism exhibiting positively expansive measures has positive topological entropy, and its restriction to thenonwandering set has the shadowingproperty. Based on this,we consider the shadowing property and entropy for the positively weak measure expansive homeomorphisms and ows, respectively. In this paper, we show that if a homeomorphism (or ow) has a positively weak expansive measure and the shadowing property on its nonwandering set, then its topological entropy is positive. This is a slight generalizationof themain result in [4].Wealso consider a relationshipbetween theweakmeasure expansivity with shadowing property and topological entropy.


Introduction
The main goal of the study on dynamical systems is to understand the structure of the orbits for homeomorphisms or ows on a compact metric space. To describe the dynamics on the underlying space, it is common to study the dynamic properties such as shadowing property, expansiveness, entropy, etc. It has close relations with stable or chaotic and sensitive properties of a given system.
Recently, Morales [1] has introduced the notion of measure expansiveness, generalizing the concept of expansiveness, and Lee et al. [2] has introduced a notion of weak measure expansiveness for ows which is really weaker than measure expansive ows in [3]. The concept of positively measure-expansiveness is introduced by [1] as a generalization of the notion of positively expansiveness, and positively measure expansive continuous maps of a compact metric space are studied from the measure theoretical point of view. Also Morales [4] proved that every homeomorphism exhibiting positively expansive measures has positive topological entropy, and its restriction to the nonwandering set has the shadowing property. Based on this, we consider the shadowing property and entropy for the positively weak measure expansive homeomorphisms and ows, respectively.
In this paper, we show that if a homeomorphism (or ow) has a positively weak expansive measure and the shadowing property on its nonwandering set, then its topological entropy is positive. This is a slight generalization of the main result in [4]. We also consider a relationship between the weak measure expansivity with shadowing property and topological entropy.

. Basics for positively weak measure expansive homeomorphisms
As pointed out by Morales [1], a notion generalizing the concept of expansiveness is called measure expansiveness. Lee et al. [2] introduced a notion of weak measure expansive homeomorphism which is weaker than the notion of measure expansive homeomorphism. From this, we study the various properties of weak measure expansive homeomorphisms, such as sensitivity, equicontinuity, shadowing property, and topological entropy.
Let (X, d) be a compact metric space and f be a homeomorphism on X. A homeomorphism f ∶ X → X is called expansive if there is δ > such that for any distinct points x, y ∈ X there exists i ∈ Z such that d(f i (x), f i (y)) > δ. Given x ∈ X and δ > , we de ne the dynamic δ-ball of f at x, Let β be the Borel σ-algebra on X. Denote by M(X) the set of Borel probability measures on X endowed with weak * topology. Let M * (X) = {µ ∈ M(X) ∶ µ be nonatomic}. A homeomorphism f ∶ X → X is said to be µ-expansive if there is δ > (called an expansive constant of µ with respect to f ) such that µ(Φ δ (x)) = for all x ∈ X. In the case, we say that f has expansive measure µ.
Now we rst introduce the notions of a nite partition P of X and a dynamical P-ball of a homeomorphism f on X. We say that a nite collection and it is called by the dynamical P-ball of f centered at x, where P(x) denotes the element of P containing x.

De nition 1.1.
A homeomorphism f on X is said to be weak µ-expansive (µ ∈ M(X)) if there exists a constant δ > and nite δ-partition P = {A , A , . . . , A n } of X such that We say that f is weak measure expansive if f is weak µ-expansive for all µ ∈ M * (X). In the case, we say that f has weak expansive measure µ.
We can also de ne the positively weak measure expansiveness for homeomorphisms by de ning the positive dynamical P-ball Γ P (x) = {y ∈ X ∶ f n (y) ∈ P(f n (x)) for all n ∈ N ∪ { }}.

De nition 1.2.
A homeomorphism f on X is said to be positively weak µ-expansive (µ ∈ M(X)) if µ(Γ P (x)) = for all x ∈ X. We say that f is positively weak measure expansive if f is positively weak µ-expansive for all µ ∈ M * (X). In the case, we say that f has positively weak expansive measure µ.
It follows easily from the de nitions that any weak measure expansive homeomorphism f is positively weak measure expansive.
We give some de nitions and notations for our works. Recall that (X, d) is a compact metric space and f ∶ X → X is a homeomorphism. The f -orbit {x, f (x), f (x), ⋯} of a point x ∈ X is denoted by O f (x). The ω-limit set ω f (x) of a point x ∈ X is the set of limit points of O f (x). We say that a point x ∈ X is periodic if f n (x) = x for some n ∈ N, recurrent if there exists n ∈ N such that f n (x) ∈ U for any neighborhood U and V of x, and non-wandering if there exists n ∈ N such that U ∩ f −n (V) ≠ ∅ for any neighborhood U of x. Let P(f ), R(f ) and Ω(f ) denote the sets of periodic, recurrent, and non-wandering points of f , respectively. Then we have A point x ∈ X is a sensitive point if there is > with the property that for any neighborhood U of x, we have diam[f n (U)] > for some n ∈ N. Let Sen(f ) denote the set of sensitive points of f . We say that f is sensitive if Sen(f ) = X and if there is > that works for all x. By the compactness of X, we see that Sen(f ) = ∅ if and only if for any > there is δ > such that d(f n (x), f n (y)) < for all n ∈ Z whenever x, y ∈ X with d(x, y) < δ. If this condition holds, we say f is equicontinuous. If x ∉ Sen(f ) then we say that f is equicontinuous at x, or x is an equicontinuity point for f .
By the compactness of X, f has shadowing property if and only if for every > there is δ > such that every nite δ-pseudo orbit Let us recall the topological entropy for a homeomorphism f on a closed set( [6]). Let n ∈ N, > , and K be a compact subset of X.
And let s n ( , K) denote the largest cardinality of any (n, )-separated subset of K with respect to f . Put So, topological entropy of f on K is de ned as the number The topological entropy of f on X is de ned as

. Basics for positively weak measure expansive flows
Many dynamic results for homeomorphisms can be extended to the case of vector elds, but not always. Bowen and Walters [5], inspired by the notion of expansiveness for discrete dynamical systems, introduced a de nition of expansiveness for continuous ows. Studying the dynamics of expansive continuous ows (or vector elds) is challenging. In this section, we begin to study the expansive ows from the measure theoretical view point.
x ∈ X and s, t ∈ R. For convenience, we will denote by The set φ R (x) is called by the orbit of φ through x ∈ X and will be denoted by O φ (x).
Let M(X) be the set of all Borel probability measures µ on X, and denote by M * φ (X) the set of µ in M(X) vanishing along the orbits of the ow φ on X. More precisely, we let More general extension, which is called measure expansivity for ows using Borel measures on a compact metric space, was introduced by Carrasco-Olivera et al. in [3]. For any ow φ on X, x ∈ X and δ > , we denote In the case, we say that φ has expansive measure µ. Now we recall that the notions of a nite δ-partition P of X and a dynamical P-ball of a homeomorphism f on X as before. For a ow φ on X, a nite δ-partition P of X and x ∈ X, the dynamical P-ball of φ centered at where H denotes the set of increasing continuous maps h ∶ R → R with h( ) = and P(x) denotes the element of P containing x.

De nition 1.3.
A ow φ on X is said to be weak µ-expansive (µ ∈ M(X)) if there exists a nite δ-partition P of X such that µ(Φ φ P (x)) = for all x ∈ X. We say that φ is weak measure expansive if φ is weak µ-expansive for all µ ∈ M * φ (X). In the case, we say that φ has weak expansive measure µ.
We can also de ne the positively weak measure expansiveness for ows by de ning the positive dynamical P-ball

De nition 1.4.
A ow φ on X is said to be positively weak µ-expansive (µ ∈ M(X)) if there exists a nite δ-partition P of X such that We say that φ is positively weak measure expansive if φ is positively weak µ-expansive for all µ ∈ M * φ (X). In the case, we say that φ has positively weak expansive measure µ.
Similarly, we can de ne periodic, recurrent, non-wandering and sensitive points for ows. A point x ∈ X is called nonwandering if for any neighborhood U of x, there is T > such that for all t ≥ T φ t (U) ∩ U ≠ ∅. The set of all nonwandering points of φ t is called the nonwandering set of φ t , denoted by Ω(φ). By non-trivial recurrence of a ow φ on a compact metric space X we mean a non-periodic point x which is recurrent in the sense that x ∈ ω(x ), where ω(x) = {y ∈ X ∶ y = lim n→∞ φ t n (x) for some sequence t n → ∞} for any x ∈ X. The set of all recurrent points of φ t is called the recurrent set of φ t , denoted by R(φ).
Let φ be a continuous ow on a compact metric space X. Given real numbers δ, a > , we say that a nite (δ, a)-chain, is a pair of sequences {(x i , t i ) ∶ i = , . . . , k} such that t i ≥ a and d(φ t i (x i ), x i+ ) < δ. An in nite (δ, a)-chain is a pair of doubly in nite sequences {(x i , t i ) ∶ i ∈ Z} such that t i ≥ a and d(φ t i (x i ), x i+ ) < δ for all i ∈ Z. The de nition of a nite(in nite) (δ, a)-pseudo orbit is the same as that of a nite(in nite) (δ, a)chain. According to standard notation let for i = , , ⋯. For every a > , the ow φ on X has the shadowing property (or pseudo-orbit tracing property) with respect to time a > if and only if φ has the shadowing property (that is with respect to time 1). For a ow φ, given any φ-invariant probability measure µ on X, we denote by h µ (φ) the measure theoretic entropy of φ with respect to µ. The topological entropy, denoted by h top (φ), can be de ned using the variational principle [9] by ; The topological entropy is always non-negative and nite.
For E, F ⊂ X we say E is a (t, δ)-separate subset of F with respect to φ if for any x, y ∈ E with x ≠ y we have max ≤s≤t d(φ s (x), φ s (y)) > δ.
Let s t (F, δ) = s t (F, δ, φ) denote the maximum cardinality of a set which is a (t, δ)-separated subset of F. If F is compact then [9] shows that s t (F, δ) < ∞. We de nē

Main Theorems . Topological entropy for positively weak measure expansive homeomorphisms
Before we state the main theorems, we recall some results from [1] and [7]. Given a map f ∶ X → X, x ∈ X, δ > and n ∈ N, we de ne for every x ∈ X, δ > , every Borel probability measure µ of X, and every sequence k l → ∞.
Based on this, we can construct the weak measure expansive set, and we will use the set for the proof of the main theorems. Let Given a measure µ ∈ M * (X) and a homeomorphism f ∶ X → X, we denote f * (µ) the pullback measure of µ denoted by f * (µ)(A) = µ(f − (A)) for all Borel set A of X. We say that a Borel measure is invariant Proof. By the de nition of Γ P (x), we can check that So we show that if µ(Γ P (x)) = then µ(Γ P (f − (x))) = for all x ∈ X, by (i) and (ii).

Lemma 2.2. Let f ∶ X → X be a homeomorphism of a metric space X. Then every invariant measure of f which is the limit with respect to weak * topology of a sequence of µ with a common expansivity constant is positively weak expansive.
Proof. As in the proof of Lemma 7 in [4], we let δ x and W[x, n]. Then we can check that for all x ∈ X, n ∈ N. Similarly, we verify that by the above fact of ( * ). So, µ is positively weak expansive measure. Since X is compact there is a subsequence µ n k such that µ n k → µ as n k → ∞. Since µ is invariant for f − and f are homeomorphisms, we have that µ is also an invariant measure of f . So, we conclude that µ is a positively weak expansive measure of f , by applying Lemma 2.2.
From the above facts, we can state the rst main theorem as following.

Theorem A.
If a homeomorphism f on X has a positively weak expansive measure and the shadowing property on its nonwandering set, then its topological entropy is positive.
The following lemma is a particular case of Corollary 6 in [8].

Lemma 2.4. If f is a homeomorphism with the shadowing property of a compact metric space X and h(f ) = ,
Proof. See Lemma 9 in [4].
Lemma 2.5. Let f ∶ X → X be a continuous map having the shadowing property on a compact metric space X. Let Y ⊂ X be an f -invariant closed set, g = f Y , and consider g in Y . If g is not equicontinuous then h(f ) > .
Proof. It is easy to prove this lemma from the next section Lemma 2.8. For more details, see Theorem 3 in [8].
We know that if h(f ) = and f has the shadowing property, then Ω(f ) is totally disconnected and f Ω(f ) ∶ Ω(f ) → Ω(f ) is an equicontinuous map. That is, an equicontinuous map of a compact metric space has zero topological entropy (for more details, Corollary 6 in [8]). The following lemma improves this result. First of all, let M * f (X) = {µ ∈ M * (X) ∶ µ be f -invariant}.
Lemma 2.6. Let f ∶ X → X be positively weak µ-expansive. Then f is not equicontinuous.
Proof. Let f be a homeomorphism of a compact metric space X. Suppose that f is equicontinuous. Since f is weak µ-expansive, there exist δ > and a nite δ-partition P = {A i ∶ i = , ⋯, n} such that µ(Γ P (x)) = for all x ∈ X. By the de nition of equicontinuous, we obtain This is a contradiction which completes the proof.
End of the Proof of Theorem A. Suppose that f is positively weak µ-expansive but h(f ) = . Then by Lemma 2.4, f Ω(f ) is equicontinuous. By Lemma 2.6, f is not positively weak measure expansive. This is a contradiction which completes the proof.

Example 2.7.
It is well-known that the horseshoe map has the shadowing property, expansive property and positive topological entropy. If a map is expansive then it has positively weak expansive measure. That is, the horseshoe map has positively weak expansive measure. So, we can conclude that this map is an example of applying Theorem A.

. Topological entropy for positively weak measure expansive flows
Let X and M * φ (X) be as before. We consider that weak measure expansive ows with the shadowing property is an extension for ows of the Theorem A.
Theorem B. If a ow φ has a positively weak expansive measure and the shadowing property on its nonwandering set, then its topological entropy is positive. Now we consider a relationship between equicontinuity and topological entropy for a ow. We say that a ow φ is equicontinuous if for any > there is δ > such that for any y ∈ X if d(x, y) < δ then d(φ t (x), φ t (y)) < for all t ∈ R. Lemma 2.8. Let X be a compact metric space and φ ∶ X × R → X be a continuous ow having the nite shadowing property. Let Y ⊂ X and ψ = φ Y . If ψ is not equicontinuous then φ has positive topological entropy.
Proof. Since ψ is not equicontinuous, there exist z ∈ Sen(ψ) with (z, z) ∈ int[R(ψ × ψ)]. Let U be a neighborhood of z in X. We have to show that h(φ, U) > . Choose > with B(z, ) ⊂ U by taking small enough. We may also assume that for any neighborhood V of z in X, there exists t ∈ R with diam[ψ t (V ∩ Y)] > . Using the shadowing property of φ, choose δ ∈ ( , ) so that every (δ, )-pseudo orbit in X is -traced by some point in X. Since It is enough to show that s t (U, δ, φ) ≥ n , and we take t = , for simplicity. For every δ ∈ ( , ) and all t ∈ R, s t (U, δ, φ) is the maximum cardinality of (U, δ, φ)-separated set for φ. Let Also, there is j ∈ N with d(φ t j (x ), φ t j (y )) > . Since we can take C = C ⋯C n ∈ {A, B} n for any n ∈ N. Then C is a δ-pseudo orbit for φ consisting of n -elements. For C ∈ {A, B} n let w C ∈ X be a point -tracing the δ-pseudo orbit C. If y ∈ {x , y } ⊂ V is the starting element of C then d(z, w C ) ≤ d(z, y) + d(y, w C ) < δ + .
Proof. Let φ be an equicontinuous ow of a compact metric space X. Suppose by contradiction that φ is a weak µ-expansive for any µ ∈ M * φ (X). Then there exist a constant δ ′ > and a nite δ ′ -partition P = {A i ∶ i = , . . . , n} of X such that µ(Γ φ P (x)) = . Letting it in the de nition of the equicontinuity, we obtain δ > (δ < δ ′ ) such that B[x, δ] ⊂ Γ φ P (x) for any x ∈ X. From this, we get µ(B[x, δ]) = for any x ∈ Ω(φ). Since X is compact, so there are nitely many points x , x , ⋯, x n such that X = ⋃ This is a contradiction, so we complete the proof.
The following lemma is an extension for a ow case of Corollary 6 in [8].
Lemma 2.10. If φ is a ow with the shadowing property on a compact metric space X and h(φ) = , then φ Ω(φ) is equicontinuous.
Proof. By Lemma 2.8, we know that if φ is weak measure expansive then φ is not equicontinuous. By Lemma 2.9, if φ is not equicontinuous then φ has positive topological entropy.
Finally, we can see that a equicontinuous positively weak measure expansive ow of a compact metric space has zero topological entropy.
End of the Proof of Theorem B. Suppose that φ is positively weak µ-expansive (µ ∈ M * φ (X)) but h(φ) = . Then by Lemma 2.10, φ Ω(φ) is equicontinuous, and so by Lemma 2.9, φ is not positively weak measure expansive. This is a contradiction.