Topological entropy for positively weak measure expansive shadowable maps

Manseob Lee; Jumi Oh

doi:10.1515/math-2018-0046

Article Open Access

Topological entropy for positively weak measure expansive shadowable maps

Manseob Lee and Jumi Oh

Published/Copyright: May 10, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

In this paper, we consider positively weak measure expansive homeomorphisms and flows with the shadowing property on a compact metric space X. Moreover, we prove that if a homeomorphism (or flow) has a positively weak expansive measure and the shadowing property on its nonwandering set, then its topological entropy is positive.

Keywords: Shadowing property; Expansive; Positively measure expansive; Positively weak measure expansive; Topological entropy

MSC 2010: 37C50; 37D20; 37B40

1 Introduction

The main goal of the study on dynamical systems is to understand the structure of the orbits for homeomorphisms or flows 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 flows which is really weaker than measure expansive flows 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 flows, respectively.

In this paper, we show that if a homeomorphism (or flow) 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.

1.1 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 > 0 such that for any distinct points x, y ∈ X there exists i ∈ ℤ such that d(fⁱ(x), fⁱ(y)) > δ. Given x ∈ X and δ > 0, we define the dynamic δ-ball of f at x,

Φδf(x)={y∈X:d(fi(x),fi(y))≤δforalli∈Z}.

(Denote Φ_δ(x) by Φδf(x) for simplicity if there is no confusion.) Then we see that f is expansive if there is δ > 0 such that Φ_δ(x) = {x} for all x ∈ 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 > 0 (called an expansive constant of μ with respect to f) such that μ(Φ_δ(x)) = 0 for all x ∈ X. In the case, we say that f has expansive measureμ. Note that Φ_δ(x) = ∩_i∈ℤf^–i (B[fⁱ(x),δ]), where B[x, δ] = {y ∈ X : d(x, y) ≤ δ}.

Now we first introduce the notions of a finite partition P of X and a dynamical P-ball of a homeomorphism f on X. We say that a finite collection P = {A₁, A₂,…, A_n} of subsets of X is a finite δ-partition (δ > 0) of X if each A_i is disjoint, measurable, intA_i ≠ ∅, diam A_i ≤ δ, and ∪i=1nAi=X. For a homeomorphism f on X, a finite δ-partition P of X and x ∈ X, we denote ΦPf(x) by

ΦPf(x)={y∈X:fi(y)∈P(fi(x))foralli∈Z},

and it is called by the dynamical P-ball of f centered at x, where P(x) denotes the element of P containing x. Denote Φ_P (x) by ΦPf(x) for simplity if there is no confusion. Then it is easy to check that Φ_P(x) is measurable, ΦP(x)=∩i=1nf−i(P(fi(x))), and f(Φ_P(x)) ⊂ Φ_P(f(x)).

Definition 1.1

A homeomorphism f on X is said to be weak μ-expansive (μ ∈ 𝓜(X)) if there exists a constant δ > 0 and finiteδ-partition P = {A₁, A₂, …,A_n} of X such that

μ(ΦPf(x))=0forallx∈X.

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 define the positively weak measure expansiveness for homeomorphisms by defining the positive dynamical P-ball

ΓP(x)={y∈X:fn(y)∈P(fn(x))foralln∈N∪{0}}.

Definition 1.2

A homeomorphism f on X is said to be positively weak μ-expansive (μ ∈ 𝓜(X)) if μ(Γ_P(x)) = 0 for all x ∈ X. We say that f is positively weak measure expansive if f is positively weakμ-expansive for allμ ∈ 𝓜^*(X). In the case, we say thatf has positively weak expansive measureμ.

It follows easily from the definitions that any weak measure expansive homeomorphism f is positively weak measure expansive.

We give some definitions 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), f²(x), …} of a point x ∈ X is denoted by 𝓞_f(x). The ω-limit setω_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 fⁿ(x) = x for some n ∈ ℕ, recurrent if there exists n ∈ ℕ such that fⁿ(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

P(f)⊂R(f)⊂Ω(f).

A point x ∈ X is a sensitive point if there is ϵ > 0 with the property that for any neighborhood U of x, we have diam[fⁿ(U)] > ϵ 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 ϵ > 0 that works for all x. By the compactness of X, we see that Sen(f) = ∅ if and only if for any ϵ > 0 there is δ > 0 such that

d(fn(x),fn(y))<ϵ

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.

For δ > 0, a δ-pseudo orbit of f in X is a finite or infinite sequence of points {xn}n=0p such that

d(xn+1,f(xn))<δ

for p ∈ ℕ ∪ {∞} and every n < p. We say that a δ-pseudo orbit {xn}n=0p is ϵ-traced by a point y ∈ X if d(x_n, fⁿ(y)) < ϵ for every n < p. And f is said to have the shadowing property if for every ϵ > 0 there is δ > 0 such that every infinite δ-pseudo orbit {xn}n=0∞ of f in X is ϵ-traced by some point in X. By the compactness of X, f has shadowing property if and only if for every ϵ > 0 there is δ > 0 such that every finite δ-pseudo orbit {xn}n=0p(p ∈ ℕ) of f in X is ϵ-traced by some point in X.

Let us recall the topological entropy for a homeomorphism f on a closed set([6]). Let n ∈ ℕ, ϵ > 0, and K be a compact subset of X. A subset E of K is said to be (n, ϵ)-separated with respect to f, if x ≠ y ∈ E implies

max0≤i≤nd(fi(x),fi(y))>ϵ.

And let s_n(ϵ, K) denote the largest cardinality of any (n, ϵ)-separated subset of K with respect to f. Put

s(ϵ,K,f)=lim supn→∞1nlogsn(ϵ,K).

So, topological entropy of f on K is defined as the number

h(f,K)=limϵ→0lim supn→∞logsn(ϵ,K)n.

The topological entropy of f on X is defined as h(f) = h(f, X). We say that x ∈ X is an entropy point for f if h(f, U) > 0 for any neighborhood U of x. Let Ent(f) denote the set of entropy points of f. Then Ent(f) is a closed f-invariant set and Ent(f) ≠ ∅ if and only if h(f) > 0.

1.2 Basics for positively weak measure expansive flows

Many dynamic results for homeomorphisms can be extended to the case of vector fields, but not always. Bowen and Walters [5], inspired by the notion of expansiveness for discrete dynamical systems, introduced a definition of expansiveness for continuous flows. Studying the dynamics of expansive continuous flows (or vector fields) is challenging. In this section, we begin to study the expansive flows from the measure theoretical view point.

Let (X, d) be a compact metric space. A flow on X is a continuous map ϕ : X×ℝ ⟶ X satisfying ϕ(x, 0) = X and ϕ(ϕ(x, s),t) = ϕ(x, s + t) for x ∈ X and t ∈ ℝ. For convenience, we will denote by

ϕ(x,s)=ϕs(x)andϕ(a,b)(x)={ϕt(x):t∈(a,b)}.

The set ϕ_ℝ(x) is called by the orbit of ϕ throughx ∈ 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 Mϕ∗(X) the set of μ in 𝓜(X) vanishing along the orbits of the flow ϕ on X. More precisely, we let

Mϕ∗(X)={μ∈M(X):μ(Oϕ(x))=0forallx∈X}.

Then we have Mϕ∗(X) ⊂ 𝓜(X). For any subset B ⊂ X (Borel measurable or not) we write μ(B) = 0 if μ(A) = 0 for any Borel subset A ⊂ B.

More general extension, which is called measure expansivity for flows using Borel measures on a compact metric space, was introduced by Carrasco-Olivera et al. in [3]. For any flow ϕ on X, x ∈ X and δ > 0, we denote Φδϕ(x)

Φδϕ(x)={y∈X:d(ϕt(x),ϕh(t)(y))≤δforsomeh∈Handallt∈R}

and it is called by the dynamical δ-ball of ϕ centered at x ∈ X. Note that

Γδϕ(x)=⋃h∈H⋂t∈Rϕ−h(t)(B[ϕt(x),δ]).

For any μ ∈ 𝓜(X), we say that ϕ is μ-expansive if there exists a constant δ > 0 such that μ(Φδϕ(x))=0 for all x ∈ X. Such a δ is called an expansivity constant of ϕ with respect to μ. Note that if ϕ is μ-expansive for some μ ∈ 𝓜(X) then μ vanishes along the orbits of ϕ (see (A4) of Theorem 1 in [3]). A flow ϕ on X is said to be measure expansive if ϕ is μ-expansive for any μ ∈ Mϕ∗(X). In the case, we say that ϕ has expansive measure μ.

Now we recall that the notions of a finite δ-partition P of X and a dynamical P-ball of a homeomorphism f on X as before. For a flow ϕ on X, a finite δ-partition P of X and x ∈ X, the dynamicalP-ball of ϕ centered at x, ΦPϕ(x), is defined by

{y∈X:ϕh(t)(y)∈P(ϕt(x))forsomeh∈Handallt∈R},

where 𝓗 denotes the set of increasing continuous maps h : ℝ → ℝ with h(0) = 0 and P(x) denotes the element of P containing x.

Definition 1.3

A flow ϕ on X is said to be weak μ-expansive (μ ∈ 𝓜(X)) if there exists a finiteδ-partition P of X such that

μ(ΦPϕ(x))=0forallx∈X.

We say that ϕ is weak measure expansive if ϕ is weak μ-expansive for all μ ∈ Mϕ∗(X). In the case, we say that ϕ has weak expansive measure μ.

We can also define the positively weak measure expansiveness for flows by defining the positive dynamical P-ball

ΓPϕ(x)={y∈X:ϕh(t)(y)∈P(ϕt(x))forsomeh∈Handallt≥0}.

Definition 1.4

A flowϕ on X is said to be positively weak μ-expansive (μ ∈ 𝓜(X)) if there exists a finiteδ-partition P of X such that

μ(ΓPϕ(x))=0forallx∈X.

We say that ϕ is positively weak measure expansive if ϕ is positively weak μ-expansive for all μ ∈ Mϕ∗(X). In the case, we say thatϕ has positively weak expansive measure μ.

Similarly, we can define periodic, recurrent, non-wandering and sensitive points for flows. A point x ∈ X is called nonwandering if for any neighborhood U of x, there is T > 0 such that for all t ≥ Tϕ_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 flow ϕ on a compact metric space X we mean a non-periodic point x₀ which is recurrent in the sense that x₀ ∈ ω(x₀), where

ω(x)={y∈X:y=limn→∞ϕtn(x)forsomesequencetn→∞}

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 flow on a compact metric space X. Given real numbers δ, a > 0, we say that a finite (δ, a)-chain, is a pair of sequences {(x_i, t_i) :i = 0, …, k} such that t_i ≥ a and d(ϕ_{t_i}(x_i), x_i+1) < δ. An infinite (δ, a)-chain is a pair of doubly infinite sequences {(x_i, t_i) : i ∈ ℤ} such that t_i ≥ a and d(ϕ_{t_i}(x_i), x_i+1) < δ for all i ∈ ℤ. The definition of a finite(infinite) (δ, a)-pseudo orbit is the same as that of a finite(infinite) (δ, a)-chain. According to standard notation let

s0=0,sn=∑i=0n−1ti,ands−n=∑i=−n−1ti

for every sequence {t_i : i ∈ ℤ} of real numbers.

Let ϵ > 0 be given.A reparametrization h ∈ 𝓗̃ satisfying h : ℝ → ℝ is a monotone increasing homeomorphism with h(0) = 0 and

|h(s)−h(t)s−t−1|≤ϵforeverys,t∈R.

A finite(infinite) (δ, a)-pseudo orbit {(x_i, t_i) : i ∈ ℤ} is ϵ-traced by an orbit (ϕ_t(z))_{t∈ℝ, z∈X} if there exists h ∈ 𝓗̃ such that

d(ϕh(t)(z),ϕt−si(xi))<ϵ,ift≥0,si≤t<si+1,d(ϕh(t)(z),ϕt+s−i(x−i))<ϵ,ift≤0,−si≤t<−s−i+1.

for i = 0, 1, …. For every a > 0, the flow ϕ on X has the shadowing property (or pseudo-orbit tracing property) with respect to time a > 0 if and only if ϕ has the shadowing property (that is with respect to time 1).

For a flow 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 defined using the variational principle [9] by ;

htop(ϕ)=sup{hμ(ϕ):μis aϕ−invariant probability measure}.

The topological entropy is always non-negative and finite.

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

max0≤s≤td(ϕs(x),ϕs(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 define

s¯ϕ(F,δ)=lim supt→∞1tlogst(F,δ)

and topological entropy by

h(ϕ,F)=limδ→0s¯ϕ(F,δ).

By Lemma 1 in [6] these limits exists and are equal. The topological entropy of ϕ is defined as h(ϕ) = h(ϕ, X). We say that x ∈ X is an entropy point for ϕ if h(ϕ, U) > 0 for any neighborhood U of x. Denote by Ent(ϕ) the set of entropy points of ϕ. Then Ent(ϕ) is a closed ϕ-invariant set and Ent(ϕ) ≠ ∅ if and only if h(ϕ) > 0.

2 Main Theorems

2.1 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 > 0 and n ∈ ℕ, we define

V[x,n,δ]={y∈X:d(fi(x),fi(y))≤δforall0≤i≤n}.

That is, V[x,n,δ]=⋂i=0nf−i(B[fi(x),δ]), where B[x, δ] denotes the closed δ-ball centered at x. It is clear that Γ_δ(x) = ⋂_n∈ℤV[x, n, δ] and V[x, m, δ] ⊂ V[x, n, δ] for n ≤ m. Consequently, μ(Γδ(x))=liml→∞μ(V[x,kl,δ]) for every x ∈ X, δ > 0, 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

VP[x,n,δ]={y∈X:fi(y)∈P(fi(x))forall0≤i≤n}

then V_P[x, n, δ] = ⋂i=0nf−i(P(fi(x))) and Γ_P(x) = ⋂_i∈ℕf^–i(P(fⁱ(x))). Similarly,

μ(ΓP(x))=liml→∞μ(VP[x,kl,δ])⋯⋯(∗).

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^–1(A)) for all Borel set A of X. We say that a Borel measure is invariant for f if μ = μ ∘ f^–1.

Lemma 2.1

Let f : X → X be a homeomorphism of a compact metric space X. If μ ∈ 𝓜^*(X) is a positively weak expansive measure with expansive constant δ of f, then so doesf_*^–1μ

Proof

By the definition of Γ_P(x), we can check that

f(Γ_P(x)) ⊂ Γ_P(f(x)) and (ii) Γ_P(x) ⊂ Γ_P(f^–1(x)).

So we show that if μ(Γ_P(x)) = 0 then μ(Γ_P(f^–1(x))) = 0 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

VP[x,n,δ2]⊂W[x,n]⊂VP[x,n,δ]

for all x ∈ X, n ∈ ℕ. Similarly, we verify that

limn→∞infμ(VP[x,n,δ2])=0

by the above fact of (*). So, μ is positively weak expansive measure. □

Lemma 2.3

If a homeomorphism f of a compact metric space X has positively weak expansive measure then it has positively weak expansive invariant measures.

Proof

Let μ be a positively weak expansive measure with expansive constant δ of f : X → X. By Lemma 2.1, we know that f_*^–1μ is a positively weak expansive measure with positive expansive constant δ of f. And so, f_*^–iμ is a positively weak expansive measure with positively expansive constant δ of f for all i ∈ ℕ, we can consider a sequence of positively weak expansive measures with uniform expansive constant δ,

μn=1n∑i=0n−1f∗−iμ,foralln∈N.

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^–1 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 first 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) = 0, then f|_Ω(f)is equicontinuous.

Proof

See Lemma 9 in [4]. □

Lemma 2.5

Let f : X → Xbe 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) > 0.

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) = 0 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 Mf∗(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 > 0 and a finite δ-partition P = {A_i : i = 1, …, n} such that μ(Γ_P(x)) = 0 for all x ∈ X. By the definition of equicontinuous, we obtain δ′ > 0 such that B[x, δ′] ⊂ Γ_P(x) for all x ∈ X. From this, we get μ(B[x, δ′]) = 0 for all x ∈ Ω(f). Since X is compact, there are finitely many points x₁, x₂, …, x_n such that X=⋃i=1nB[xn,δ′]. Then

μ(X)≤∑i=1nμ(B[xi,δ′])=0.

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) = 0. 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 applyingTheorem A.

2.2 Topological entropy for positively weak measure expansive flows

Let X and Mϕ∗(X) be as before. We consider that weak measure expansive flows with the shadowing property is an extension for flows of the Theorem A.

Theorem B

If a flow 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 flow. We say that a flow ϕ is equicontinuous if for any ϵ > 0 there is δ > 0 such that for any y ∈ X if d(x, y) < δ then d(ϕ_t(x), ϕ_t(y)) < ϵ for all t ∈ ℝ.

Lemma 2.8

Let X be a compact metric space and ϕ : X × ℝ → X be a continuous flow having the finite shadowing property. Let Y ⊂ X andψ = 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) > 0. Choose ϵ > 0 with B(z, 2ϵ) ⊂ U by taking ϵ 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)] > 3ϵ. Using the shadowing property of ϕ, choose δ ∈ (0, ϵ) so that every (δ, 1)-pseudo orbit in X is ϵ-traced by some point in X.

Since V is a neighborhood of z in X with (V ∩ Y) × (V ∩ Y) ⊂ int[R(ψ × ψ)] and diam(V) < δ2. Then there exist T ∈ ℝ and (x₀, y₀) ∈ R(ψ × ψ) ∩ (V × V) such that d(ψ_T(x₀), ψ_T(y₀)) > 3ϵ. Since (x₀, y₀) ∈ R(ψ × ψ), there is τ > T with

d(x0,ψτ(x0))<δ2andd(y0,ψτ(y0))<δ2.

Now we claim that

h(ϕ,U¯)≥log2τ.

It is enough to show that s_t(U, δ, ϕ) ≥ 2ⁿ, and we take t = 1, for simplicity. For every δ ∈ (0, ϵ) and all t ∈ ℝ, s_t(U, δ, ϕ) is the maximum cardinality of (U, δ, ϕ)-separated set for ϕ. Let

A={(xi,ti):ti≥1,i=0,1,⋯,n−1}andB={(yi,ti):ti≥1,i=0,1,⋯,n−1}

satisfying

d(ϕti(xi),xi+1)<δ,foranyi=0,1,⋯,n−1andd(ϕti(yi),yi+1)<δ,foranyi=0,1,⋯,n−1.

Also, there is j ∈ ℕ with d(ϕ_{t_j}(x₀), ϕ_{t_j}(y₀)) > 3ϵ. Since

d(x0,y0)<δ2,d(x0,ψtj(x0))<δ2andd(y0,ψtj(x0))<δ2,

we can take C = C₁ … C_n ∈ {A, B}ⁿ for any n ∈ ℕ. Then C is a δ-pseudo orbit for ϕ consisting of n²-elements. For C ∈ {A, B}ⁿ let w_C ∈ X be a point ϵ-tracing the δ-pseudo orbit C. If y ∈ {x₀, y₀} ⊂ V is the starting element of C then

d(z,wC)≤d(z,y)+d(y,wC)<δ+ϵ.

So, w_C ∈ U. If C, D ∈ {A, B}ⁿ are distinct then for some k ∈ {0, 1, …, n – 1}, the k-th elements of the pseudo orbits C and D are more than 3ϵ apart. Therefore by the triangle inequality,

d(ϕtk(wC),ϕtk(wD))>ϵforsomek∈{0,1,⋯,n−1}.

This means that the set {w_C : C ∈ {A, B}ⁿ} is (U, ϵ, ϕ)-separated and hence (U, δ, ϕ)-separated for ϕ and for any δ ∈ (0, ϵ). That is,

|{wC:C∈{A,B}n}|=2n.

So, we complete the proof. □

Now we introduce the notion of V_P[ϕ, x, T, δ] which is a flow case of V_P[x, n, δ]. Let V_P[ϕ, x, T, δ] = {y ∈ X : ϕ_h(t)(y) ∈ P(ϕ_t(x)) for some h ∈ 𝓗̃ and –T ≤ t ≤ T}. Then

VP[ϕ,x,T,δ]=⋃h∈H~⋂−T≤t≤Tϕ−h(t)(B[ϕt(x),δ]).

Similarly, μ(ΓPϕ(x))=limt→∞μ(VP[ϕ,x,t,δ]).

Lemma 2.9

Let ϕ be positively weak μ-expansive. Then ϕ is not equicontinuous.

Proof

Let ϕ be an equicontinuous flow of a compact metric space X. Suppose by contradiction that ϕ is a weak μ-expansive for any μ∈Mϕ∗(X). Then there exist a constant δ′ > 0 and a finite δ′-partition P = {A_i : i = 1, …, n} of X such that μ(ΓPϕ(x))=0.

Letting it in the definition of the equicontinuity, we obtain δ > 0(δ < δ′) such that B[x, δ] ⊂ ΓPϕ(x) for any x ∈ X. From this, we get μ(B[x, δ]) = 0 for any x ∈ Ω(ϕ). Since X is compact, so there are finitely many points x₁, x₂, …, x_n such that X=⋃i=1nB[xi,δ]. Then

μ(X)≤∑i=1nμ(B[xi,δ])=0.

This is a contradiction, so we complete the proof. □

The following lemma is an extension for a flow case of Corollary 6 in [8].

Lemma 2.10

If ϕ is a flow with the shadowing property on a compact metric space X and h(ϕ) = 0, thenϕ|_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 flow 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∈Mϕ∗(X)) but h(ϕ) = 0. 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. □

Acknowledgement

The first author is supported by the National Research Foundation of Korea (NRF) No. 2017R1A2B4001892. The second author is supported by the National Research Foundation of Korea (NRF) No. 2016R1D1A1B03931962 and No. 2015R1A3A2031159.

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Received: 2017-11-28

Accepted: 2018-03-07

Published Online: 2018-05-10

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0046

Keywords for this article

Shadowing property; Expansive; Positively measure expansive; Positively weak measure expansive; Topological entropy

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