An Open Cover Specification Property

Abstract

In this paper, we define and study the topological version of the specification property, called open cover specification property. We explore its connections with different types of mixing properties, such as topological mixing, total transitivity, weak mixing, and the non-wandering property. We also establish that, in the presence of positive orbit expansivity on a compact Hausdorff space, the open cover specification property implies local eventual onto behaviour. Additionally, we calculate the topological entropy of a dynamical system on a compact Hausdorff space, showing that it can be expressed in terms of periodic points when the map satisfies both the open cover specification property and the positive orbit expansivity.

Introduction

Understanding the structure of orbits is a fundamental aspect of topological dynamics. One key concept in this field is the specification property, which was introduced by Bowen [6]. This property ensures that the true trajectories of points can be found near their approximate counterparts. Specifically, it states that any finite segment of orbits, sufficiently separated in time, can be approximately traced by some point in the space. Therefore, studying the specification property becomes crucial for gaining insight into the behaviour of trajectories. In this research paper, we define and investigate a topological version of the specification property. We explore its relationship with other important properties such as topological shadowing, topological entropy, local eventual onto (L.E.O.) etc. for general topological spaces. By studying these connections, we aim to deepen our understanding of the dynamics and structure of orbits in topological systems.

The specification property has been widely recognized by various researchers for its usefulness. Specifically, it has been proved that, on a compact metric space, the specification property and topological mixing are equivalent for a surjective map that possesses the shadowing property [18]. Moreover on a compact interval, the periodic specification property and mixing coincide [3]. The correlation between the specification property and L.E.O. has been examined [24]. Several authors have investigated the specification property in the presence of expansivity [2, 6, 11, 20, 23]. Furthermore, there have been investigations into the relationship between the specification property and the entropy of the system [4, 7, 14, 21]. Bowen discovered a connection between entropy and periodic points of an invertible expansive homeomorphism on a compact metric space when the map has periodic specification property [6]. Weaker versions of the specification property are also being explored by several authors [7, 9, 10, 18, 20, 22, 25]. In the settings of Symbolic Dynamics, a notion of weak specification has been investigated in Exercises $1.66-1.69$ [8].

In our investigations, we focus on the significance of the specification property and introduce its topological counterpart, called open cover specification property, along with a definition for positive orbit expansivity. We delve into the study of the open cover specification property, positive orbit expansivity, mixing, L.E.O., and topological entropy. One of our key results establishes that if a map on a compact Hausdorff space possesses both open cover specification and positive orbit expansivity, then the map is L.E.O. Additionally, we provide a significant result regarding topological entropy, showing that it can be expressed in terms of periodic points for a map, on a compact Hausdorff space, that exhibits the periodic open cover specification property and positive orbit expansivity. In Section 2, we present the essential definitions that are necessary for understanding the subsequent sections. In Section 3, we begin by introducing the topological versions of specification property and expansivity. We then investigate the relationship between the open cover specification properrty and mixing. We prove that, on a compact Hausdorff space in the presence of the topological shadowing property, topological mixing implies the open cover specification property. We also establish connections among various mixing properties, such as weak mixing, total transitivity, non-wandering, and the open cover specification property. Furthermore, in Section 3, we prove that on a compact Hausdorff space, any map exhibiting the open cover specification property and positive orbit expansivity is L.E.O. In Section 4, we obtain a result that establishes a relationship between open covers and periodic points on compact Hausdorff spaces in the context of a map having the periodic open cover specification property. We also prove that on a compact Hausdorff space, if a self map possess positive orbit expansivity and the open cover specification property then the topological entropy of the map can be determined in terms of the periodic points of the map. Throughout our paper, we provide relevant examples and counterexamples to support and illustrate our findings.

Preliminaries

We shall denote the set of natural numbers by $\mathbb{N}$ ($0\notin \mathbb{N}$) and the set of integers by $\mathbb{Z}$.

Let X be any topological space, for any pair of sets $A,B\subset X\times X$, the composition of A and B, denoted by $A\circ B$, is defined as $A\circ B=\{(x,z):\text{there exists a}y\in X\text{such that}(x,y)\in A\text{and}(y,z)\in B\}.$ The diagonal of $X\times X$ is defined as ${\mathrm{\Delta }}_X=\{(x,x):x\in X\}$, and the set ${\mathcal{D}}_X$ is defined as ${\mathcal{D}}_X=\{A\subset X\times X:{\mathrm{\Delta }}_X\subset A\text{and}A=A^{-1}\},$ where for any set $A\subset X\times X$, $A^{-1}$ is defined as $A^{-1}=\{(y,x):(x,y)\in A\}.$

Definition 2.1

[17] For any set X, a uniformity on X is a non-empty collection $\phi$ of subsets of $X\times X$ satisfying the following conditions:

1. $\phi \subset {\mathcal{D}}_X$.

2. If $A_1\in \phi$ and $A_1\subset A_2$, then $A_2\in \phi$.

3. For any $A_1,A_2\in \phi$, $A_1\cap A_2\in \phi$.

4. For any $A\in \phi$, there exists $B\in \phi$ such that $B\circ B\subset A$.

The pair $(X,\phi )$ is called a uniform space, and the elements of $\phi$ are called entourages.

Definition 2.2

Any uniform space $(X,\phi )$ induces a topology on X, called the uniform topology generated by basic open sets $A[x]=\{y:(x,y)\in A\}$, where $A\in \phi$. We say that the topology is induced by the uniformity $\phi$.

Let X be any topological space and $f:X\to X$ be a continuous map, the pair (X, f) is called a topological dynamical system. A point $x\in X$ is called periodic if there exists an $n\in \mathbb{N}$ such that $f^n(x)=x$. We denote $N_n(f)=|\{x\in X:f^n(x)=x\}|$ where |S| denotes the cardinality of the set S. A point $x\in X$ is called non-wandering if for any neighbourhood U of x, there exists an $n\in \mathbb{N}$ such that $f^n(U)\cap U\ne \mathrm{\varnothing }$; moreover if for any non-empty open set U of X there exists an $n\in \mathbb{N}$ such that $f^n(U)\cap U\ne \mathrm{\varnothing }$ then (X, f) is called non-wandering. The system (X, f) is called transitive if for any pair of non-empty open sets $U,V\subset X$ there exists an $n\in \mathbb{N}$ such that $f^n(U)\cap V\ne \mathrm{\varnothing }$. We say that (X, f) is totally transitive if for any $k\in \mathbb{N}$, $(X,f^k)$ is transitive. The system (X, f) is said to be weakly mixing if for any pair of non-empty open sets $U_1,U_2;V_1,V_2\subset X$, there exists an $n\in \mathbb{N}$ such that $f^n(U_1)\cap V_1\ne \mathrm{\varnothing }$ and $f^n(U_2)\cap V_2\ne \mathrm{\varnothing }$. If for any non-empty open set $U\subset X$ there exists an $n\in \mathbb{N}$ such that $f^n(U)=X$ then (X, f) is called locally eventually onto.

For any uniform space X, any entourage E and $k\in \mathbb{N}$, a set $A\subset X$ is called $(k,E)-$separated if for any $x,y\in A,x\ne y$ there exists an $i\in \{0,1,\cdots ,k\}$ such that $(f^i(x),f^i(y))\notin E$ [15].

For a natural number N, any set of integers $a_1\le b_1<a_2\le b_2<\cdots <a_n\le b_n$ with $(a_j-b_{j-1})\ge N$ for every $j\in \{2,3,\cdots ,n\}$ and any collection of points $x_1,x_2,\cdots ,x_k\in X$, the family of orbits $\xi ={\{f^{[a_j,b_j]}(x_j)\}}_{j=1}^n$, where $f^{[a_j,b_j]}(x_j)=(f^{a_j}(x_j),f^{a_j+1}(x_j),\cdots ,f^{b_j}(x_j)$), is called an N-spaced specification.

Let X be a metric space with metric d and $f:X\to X$ be a continuous map. We say that a specification $\xi ={\{f^{[a_j,b_j]}(x_j)\}}_{j=1}^n$ is $ϵ$-traced if there exists a $y\in X$ such that $d(f^k(x_j),f^k(y))\le ϵ$ for every $a_j\le k\le b_j$ and every $j\in \{1,2,\cdots ,n\}$. A dynamical system (X, f) is called positively expansive if there exists a $\beta >0$ such that for any pair of distinct $x,y\in X$ there exists an $n\in \mathbb{N}\cup \{0\}$ such that $d(f^n(x),f^n(y))>\beta$ [12]. For any topological space X a homeomorphism $f:X\to X$ is said to have topological specification property if for any symmetric neighbourhood E of the diagonal ${\mathrm{\Delta }}_X$ there exists a positive integer M such that for any $M-$spaced specification $\xi ={\{f^{[a_j,b_j]}(x_j)\}}_{j=1}^n$, there exists an $x\in X$ such that $(f^k(x),f^k(x_j))\in E$ for any $a_j\le k\le b_j$ and any $j\in \{1,2,\cdots ,n\}$ [23]. For any topological space X and any homeomorphism $f:X\to X$, the dynamical system (X, f) is called topologically expansive if there exists a closed neighbourhood E of the diagonal ${\mathrm{\Delta }}_X=\{(x,x):x\in X\}$ such that for any $x,y\in X,x\ne y$ there exists an $n\in \mathbb{Z}$ such that $(f^n(x),f^n(y))\notin E$; E is called an expansivity neighbourhood [13]. A homeomorphism $f:X\to X$ is called orbit expansive if there exists a finite open cover $\mathcal{B}$ of X such that for any $x,y\in X,x\ne y$ there exists an $n\in \mathbb{Z}$ such that $(f^n(x),f^n(y))\notin B\times B$ for any $B\in \mathcal{B}$; open cover $\mathcal{B}$ is called o-expansive cover [1].

We say that a dynamical system (X, f) has specification property if given any $ϵ>0$ there exists an $N\in \mathbb{N}$ such that any $N-$spaced specification is $ϵ$-traced by some $y\in X$. If $f^{b_n+N}(y)=y$, then (X, f) is said to have periodic specification property [6].

Definition 2.3

[19] Given any dynamical system (X, f).

1. Let $\overline{x}=(x_0,x_1,x_2,\cdots )$ be a sequence in X and $\mathcal{U}$ be an open cover of X then we say that $\overline{x}$ is a $\mathcal{U}$-pseudo orbit if for any $n\in \mathbb{N}\cup \{0\}$, $(f(x_n),x_{n+1})\in U\times U$ for some $U\in \mathcal{U}$.

2. Let $\mathcal{V}$ be any open cover of X and $\overline{y}=(y_0,y_1,y_2,\cdots )$ be a sequence in X then we say that $\overline{y}$ is $\mathcal{V}$-shadowed if there exists a $z\in X$ such that for any $n\in \mathbb{N}\cup \{0\}$, $(y_n,f^n(z))\in V\times V$ for some $V\in \mathcal{V}$.

3. The dynamical system (X, f) is said to have the topological shadowing property, if for any open cover $\mathcal{V}$, there exists an open cover $\mathcal{U}$ such that every $\mathcal{U}$-pseudo orbit is $\mathcal{V}$-shadowed.

Let X be any topological space and $\mathcal{U}$ and $\mathcal{V}$ be any open covers of X. Then, the join of $\mathcal{U}$ and $\mathcal{V}$ is defined by $\mathcal{U}\vee \mathcal{V}=\{U\cap V:U\in \mathcal{U},V\in \mathcal{V}\}.$ For a topological space X, an open cover $\mathcal{U}$ of X, and a continuous map $f:X\to X$, we define the preimage cover as $f^{-1}(\mathcal{U})=\{f^{-1}(U):U\in \mathcal{U}\}$, and inductively, for any $n\in \mathbb{N}$, $f^{-(n+1)}(\mathcal{U})=\{f^{-1}(f^{-n}(U)):U\in \mathcal{U}\}.$ Let X be any compact topological space and $\mathcal{U}$ be any open cover of X. The the minimum cardinality of any subcover of $\mathcal{U}$ that covers X is denoted by $N(\mathcal{U})$ and $H(\mathcal{U})=log(N(\mathcal{U}))$. For any compact topological space X, an open cover $\mathcal{U}$ of X, and a continuous map $f:X\to X$, the topological entropy of  f  with respect to the open cover$\mathcal{U}$ is denoted by $h(f,\mathcal{U})$ and is defined as:$$h(f,\mathcal{U})=\left\{\underset{n\to \infty }{lim},\frac{1}{n},H,(\mathcal{U}\vee f^{-1}(\mathcal{U})\vee \cdots \vee f^{-(n-1)}(\mathcal{U})),:,\mathcal{U},,\text{is an open cover of},,X\right\}.$$The topological entropy of f is denoted by h(f) and is defined as:$$h(f)=sup\left\{\underset{n\to \infty }{lim},\frac{1}{n},H,(\mathcal{U}\vee f^{-1}(\mathcal{U})\vee \cdots \vee f^{-(n-1)}(\mathcal{U})),:,\mathcal{U},,\text{is an open cover of},,X\right\}.$$

Open Cover Specification, Mixing and L.E.O.

In this section, we focus on exploring the relationship among the topological version of the specification property, mixing, and L.E.O. Shah et al. defined a topological version of specification property for homeomorphisms on a compact topological space, say X, using neighbourhoods of the diagonal in $X\times X$ [23]. We begin by introducing topological versions of specification (defined using open covers) and positive expansivity.

Definition 3.1

Given any open cover $\mathcal{V}$ of  X, we say that a specification $\xi ={\{f^{[a_j,b_j]}(x_j)\}}_{j=1}^n$ is $\mathcal{V}$-traced if there exists a $y\in X$ such that for every $a_j\le k\le b_j$, $(f^k(y),f^k(x_j))\in V\times V$ for some $V\in \mathcal{V}$ depending on k. The dynamical system (X, f) is said to have the open cover specification property if given any open cover $\mathcal{V}$ of X there exists an $N\in \mathbb{N}$ such that any $N-$spaced specification is $\mathcal{V}$-traced by some $y\in X$. If, in addition $f^{b_n+N}(y)=y$, then we say that (X, f) has the periodic open cover specification property.

Achigar et al. defined the concept of orbit expansivity using open covers for homeomorphisms on topological spaces [1]. Here, we define positive orbit expansivity for continuous maps on topological spaces.

Definition 3.2

Let (X, f) be any dynamical system and $\mathcal{B}$ be an open cover of X, we say that $\mathcal{B}$ is a positive o-cover for (X, f) if for any $x,y\in X,x\ne y$ there exists an $n\in \mathbb{N}\cup \{0\}$ satisfying $(f^n(x),f^n(y))\notin B\times B$ for any $B\in \mathcal{B}$. We say that (X, f) is positive orbit expansive if there exists an o-cover for (X, f).

In the context of continuous maps on a compact interval, it has been established that a map possesses the specification property if and only if it exhibits topological mixing [3]. Furthermore, S. Shah et al. demonstrated that a self-homeomorphism on a totally bounded uniform space, which is mixing, topologically expansive, and possesses shadowing, also possesses the topological specification property [23]. Similarly, in the case of a continuous surjective map on a compact metric space with the shadowing property, the specification property and topological mixing are equivalent [18]. Here, we extend these results to general topological spaces, providing a proof of the equivalence between the open cover specification property and topological mixing.

Proposition 3.1

Let (X, f) be any dynamical system on a $T_3$ space X with f an onto continuous map. If (X, f) has the open cover specification property then (X, f) is topological mixing.

Proof

Let U and V be any pair of non-empty open subsets of X. Take $x\in U$ and $y\in V$ and $U^{\prime },V^{\prime }$ open sets such that $x\in U^{\prime }\subset \overline{U^{\prime }}\subset U,y\in V^{\prime }\subset \overline{V^{\prime }}\subset V$. Then $\mathcal{V}=\{U,V,X\backslash (\overline{U^{\prime }}\cup \overline{V^{\prime }})\}$ is an open cover of X. As (X, f) has the open cover specification property, there exists an $N\in \mathbb{N}$ such that any $N-$spaced specification is $\mathcal{V}$ traced by some $y\in X$. Consider the specification $\xi =\{x,f^n(f^{-n}(y))\}$ where $n>N$ and $a_1=0=b_1,a_2=n=b_2$. Then there exists a $z\in X$ such that $\xi$ is $\mathcal{V}$-traced by z. So, $(x,z)\in U\times U$ and $(y,f^n(z))\in V\times V$ where $U,V\in \mathcal{V}$. Hence, for every $n>N$ we have $f^n(U)\cap V\ne \mathrm{\varnothing }$ implying that (X, f) is topologically mixing.$\square$

Proposition 3.2

Let (X, f) be any dynamical system with compact and $T_3$ space X and an onto map f. If (X, f) has the topological shadowing property and is topological mixing then (X, f) has the open cover specification property.

Proof

Let $\mathcal{V}$ be any open cover of X. As X has the topological shadowing property, there exists an open cover $\mathcal{U}$ such that any $\mathcal{U}$-pseudo orbit is $\mathcal{V}$ shadowed. Since X is compact, without loss of generality, we can assume that $\mathcal{U}$ is finite, say $\mathcal{U}=\{U_1,U_2,\cdots ,U_m\}$. Then as f is topological mixing, so there exists an $N\in \mathbb{N}$ such that for every $n\ge N$, $f^n(U_i)\cap U_j\ne \mathrm{\varnothing }$ for all $i,j\in \{1,2,\cdots ,m\}$.

Let $\xi ={\{f^{[a_i,b_i]}(x_i)\}}_{i=1}^m$ be any $(N+1)-$spaced specification. For every $i\in \{1,2,\cdots ,m\}$ take $U_{a_i},U_{b_i}\in \mathcal{U}$ containing $f^{a_i}(x_i),f^{b_i+1}(x_i)$ respectively. Then for any $i\in \{1,2,\cdots ,$$m-1\}$, $f^{c_i}(U_{b_i})\cap U_{a_{i+1}}\ne \mathrm{\varnothing }$ for $c_i=(a_{i+1}-b_i-1)\ge N$. Let $x_{b_i}\in U_{b_i}$ be an element such that $f^{c_i}(x_{b_i})\in U_{a_{i+1}}$ for any $i\in \{1,2,\cdots ,m-1\}$.

Now, consider the $\mathcal{U}$-pseudo orbit $\overline{x}=\{x_1,f(x_1),\cdots ,f^{a_1}(x_1),\cdots ,f^{b_1}(x_1),$$x_{b_1},f(x_{b_1}),$$\cdots ,$$f^{c_1-1}(x_{b_1}),f^{a_2}(x_2),\cdots ,f^{b_m}(x_m)\}$. From above paragraph, we know that $(f^{b_1+1}(x_1),x_{b_1})\in U_{b_1}\times U_{b_1},(f^{c_1}(x_{b_1}),f^{a_2}(x_2))\in U_{a_2}\times U_{a_2},(f^{b_2+1}(x_2),x_{b_2})\in U_{b_2}\times U_{b_2}$ and so on. Therefore, the orbit $\overline{x}$ is a $\mathcal{U}$ pseudo orbit. Hence, there exists a $y\in X$ such that $\overline{x}$ is $\mathcal{V}$ shadowed by $y\in X$ and therefore $\xi$ is $\mathcal{V}$-traced by $y\in X$ implying that (X, f) has the open cover specification property.$\square$

Next, we state a result relating topological mixing with topological weak mixing and the non-wandering property.

Lemma 3.1

[19] Let (X, f) be any dynamical system on a compact Hausdorff space X having the topological shadowing property. We have the following:

1. If (X, f) is weakly mixing then (X, f) is topologically mixing.

2. If (X, f) is totally transitive then (X, f) is weakly mixing.

3. If X is connected and $f:X\to X$ is non-wandering then (X, f) is totally transitive.

From Lemma above and Proposition 3.2, one can deduce the following.

Theorem 3.1

If a dynamical system (X, f), on a compact Hausdorff space X, has the topological shadowing property then the following are equivalent:

1. (X, f) has the open cover specification property.

2. (X, f) is topologically mixing.

3. (X, f) is weakly mixing.

4. (X, f) is totally transitive.

5. (X, f) is non-wandering (if X is connected also).

Example 3.1

Consider $S^1=\{e^{\iota \theta }:\theta \in \left[0,,,2,\pi \right)\}$ with usual metric and let $f:S^1\to S^1$ be defined by $f(e^{\iota \theta })=e^{2\iota \theta }$. Then f is a non-wandering map. Let $\mathcal{U}$ be any open cover of $S^1$. By compactness of $S^1$, there exists a Lebesgue number, say $ϵ$, for $\mathcal{U}$. Take $\mathcal{V}=\{B(x,ϵ/2):x\in S^1\}$. We claim that any $\mathcal{V}-$pseudo orbit is $\mathcal{U}-$shadowed by some $x\in X$. Let $y_0,y_1,\cdots y_k$ be any $\mathcal{V}-$pseudo orbit. Then $f(B(y_i,ϵ))=B(f(y_i),2ϵ)\supset B(y_{i+1},ϵ)$ for any $i\in \{0,1,\cdots k\}$ which implies that $B(y_0,ϵ)\cap f^{-1}(B(y_1,ϵ))\cap \cdots f^{-k}(B(y_k,ϵ))\ne \mathrm{\varnothing }.$ Hence, $(S^1,f)$ has the topological shadowing property and therefore by Theorem 3.1, $(S^1,f)$ has the open cover specification property.

Blokh proved that for a continuous map on a compact interval, the notions of periodic specification and topological mixing are the same [3]. Subsequently, Blokh also established a result demonstrating that mixing implies a more enhanced version of the specification property on a compact interval [5]. Here, we prove that any continuous map on a compact Hausdorff topological space with positive orbit expansivity and the open cover specification property is L.E.O.

Lemma 3.2

Let X be a $T_3$ space and $f:X\to X$ be a continuous map. Then (X, f) is positive orbit expansive if and only if there exists an open cover ${\mathcal{B}}^{\prime }$ such that for any pair of distinct points $x,y\in X$ there exists an $n\in \mathbb{N}$ satisfying $(f^n(x),f^n(y))\notin \overline{B^{\prime }}\times \overline{B^{\prime }}$ for any $B^{\prime }\in {\mathcal{B}}^{\prime }$.

Proof

Let (X, f) be positive orbit expansive and $\mathcal{B}$ be an o-over for (X, f). Then X being a $T_3$ space, for any $z\in X$, there exists an open set $B_z$ such that $z\in B_z\subset {\overline{B}}_z\subset B$ for some $B\in \mathcal{B}$. The collection ${\mathcal{B}}^{\prime }=\{B_z|z\in X\}$ forms an open cover for X and for any pair of distinct $x,y\in X$ if $(f^n(x),f^n(y))\notin B\times B$ for any $B\in \mathcal{B}$ then $(f^n(x),f^n(y))\notin \overline{B^{\prime }}\times \overline{B^{\prime }}$ for any $B^{\prime }\in {\mathcal{B}}^{\prime }$.

Conversely, suppose that there exists an open cover ${\mathcal{B}}^{\prime }$ such that for any pair of distinct points $x,y\in X$ there exists an $n\in \mathbb{N}$ satisfying $(f^n(x),f^n(y))\notin \overline{B^{\prime }}\times \overline{B^{\prime }}$ for any $B^{\prime }\in {\mathcal{B}}^{\prime }$. Since $(f^n(x),f^n(y))\notin \overline{B^{\prime }}\times \overline{B^{\prime }}$, $(f^n(x),f^n(y))\notin B^{\prime }\times B^{\prime }$. Hence, we have an open cover ${\mathcal{B}}^{\prime }$ such that for any pair of distinct points $x,y\in X$ there exists an $n\in \mathbb{N}$ satisfying $(f^n(x),f^n(y))\notin B^{\prime }\times B^{\prime }$ for any $B^{\prime }\in {\mathcal{B}}^{\prime }$. Thus (X, f) is positive orbit expansive. $\square$

Theorem 3.2

Let (X, f) be any dynamical system, on a compact Hausdorff space X, having the open cover specification property and f be a positive orbit expansive map. Then (X, f) is L.E.O.

Proof

Take any non-empty open set G and the open cover ${\mathcal{B}}^{\prime }$ as defined in the Lemma 3.2. Choose an $a\in G$ and an open set $G_1$ such that $a\in G_1$ and $G_1$ satisfies $\overline{G_1}\subset G_2\subset \overline{G_2}\subset G$. Now, consider the open cover $\mathcal{V}={\mathcal{B}}^{\prime }\vee \{G_2,X/\overline{G_1}\}$. By the open cover specification property, there exists an $N\in \mathbb{N}$ such that any $N-$spaced specification is $\mathcal{V}-$traced.

For any $k\in \mathbb{N}$, take an $N-$spaced specification $S_k=\{a,f^{[N,N+k]}(f^{-N}(y))\}$ where $y\in X$ is an arbitrary element. Let $S_k$ be $\mathcal{V}-$traced by $x_k$. Then $(f^N(x_k),y)\in V\times V$ for some $V\in \mathcal{V}$. Since X is compact therefore the sequence $\overline{x}={(f^N(x_k))}_{k\in \mathbb{N}}$ has a limit point, say z. If $z\ne y$, then by positive orbit expansivity of f, there exists an $n\in \mathbb{N}$ such that $(f^n(y),f^n(z))\notin \overline{B^{\prime }}\times \overline{B^{\prime }}$ for any $B^{\prime }\in {\mathcal{B}}^{\prime }$. For any $B_y^{\prime }\in {\mathcal{B}}^{\prime }$ containing $f^n(y)$, $f^n(z)\notin \overline{B_y^{\prime }}$. Hence, $f^n(z)\in X/\overline{B_y^{\prime }}$ i.e. $z\in f^{-n}(X/\overline{B_y^{\prime }}).$

Note that for any $k>n$, $f^N(x_k)\in V_y\cap f^{-1}(V_{f(y)})\cap f^{-2}(V_{f^2(y)})\cap \cdots \cap f^{-k}(V_{f^k(y)})$ where $V_{f^i(y)}\in \mathcal{V}$ is an open set containing $f^i(y)$ for any $i\in \{0,1,\cdots ,k\}$. As $\mathcal{V}$ is a refinement of ${\mathcal{B}}^{\prime }$ and $f^n(y)\in V_{f^n(y)}$, so $f^n(z)\notin \overline{V_{f^n(y)}}$ and hence, $z\in f^{-n}(X/\overline{V_{f^n(y)}})$. Since $\left(V_y,\cap ,f^{-1},(V_{f(y)}),\cap ,f^{-2},(V_{f^2(y)}),\cap ,\cdots ,\cap ,f^{-k},(V_{f^k(y)})\right)\bigcap \left(f^{-n},(X/\overline{V_{f^n(y)}})\right)=\mathrm{\varnothing }.$ Therefore, for any $k>n$, $f^N(x_k)\notin f^{-n}(X/\overline{V_{f^n(y)}})$ implying that z is not a limit point of the sequence ${(f^N(x_k))}_{k\in \mathbb{N}}$.

Hence, for any $z\ne y$, z is not a limit point of sequence $\overline{x}$. Since X is a compact space, we get that y is the limit of $\overline{x}$.

Let x be a limit point of the sequence ${(x_k)}_{k\in \mathbb{N}}$. Then $x\in {\overline{G}}_2\subset G$. Hence, for any $y\in X$ there exists an $x\in G$ such that $f^N(x)=y$ implying that (X, f) is L.E.O. $\square$

In the following example, we justify the necessity of the condition of positive orbit expansivity used in the hypothesis of previous result.

Example 3.2

Let $X=[0,1]$ with usual metric and $p_i=\frac{1}{2^{i+1}},\text{if}i\in \{0,-1,-2\cdots \}$ and $p_i=1-\frac{1}{2^i}$, if $i\in \{1,2,\cdots \}$. For any $i\in \mathbb{Z}$ take $I_i=[p_i,p_{i+1}]$ and define $f_i:I_i\to I_{i-1}\cup I_i\cup I_{i+1}$ by setting $f_i(p_i)=p_i,f_i(p_{i+1})=p_{i+1},f_i(\frac{2p_i+p_{i+1}}{3})=p_{i-1},f_i(p_i+2p_{i+1})=p_{i+2}$ and $f_i$ is linear on each of subintervals $[p_i,\frac{2p_i+p_{i+1}}{3}]$, $[\frac{2p_i+p_{i+1}}{3}$, $\frac{p_i+2p_{i+1}}{3}]$ and $[\frac{p_i+2p_{i+1}}{3},p_{i+1}]$.

Define $f:X\to X$ by $f(0)=1,f(1)=1,f(x)=f_i(x)$ if $x\in I_i.$ Then f is a topologically mixing map and hence, (X, f) has the open cover specification property. Since f is not one-one map, f is not positive orbit expansive and as $f^{-1}\{0\}=\{0\}$, f cannot be L.E.O either.

Open Cover Specification and Topological Entropy

In this section, our focus will be on exploring the relationship between open cover specification and topological entropy.. It is quite evident that topological entropy of a non trivial dynamical system with the periodic open cover specification property is positive(Refer [6] for the metric version of the result). For metric spaces, if a homeomorphism f is expansive and has periodic specification property then the entropy of the dynamical system is $h(f)=\underset{n\to \infty }{lim}\frac{1}{k}log(N_k(f))$ [6]. Here, we extend this result to a dynamical system (X, f) on a compact Hausdorff topological space X and any continuous map $f:X\to X$.

We recall some results.

Lemma 4.1

[16] A space is uniform if and only if it is completely regular.

Lemma 4.2

[17] Let $(X,\tau )$ be a compact uniform space with topology $\tau$ induced by uniformity $\phi$. Then every neighborhood of the diagonal in $X\times X$ is a member of $\phi$.

Theorem 4.1

Let (X, f) be a dynamical system with X a compact Hausdorff space. Suppose that (X, f) has the periodic open cover specification property and let $\mathcal{V}$ be an open cover of X. Then there exists an $N_{\mathcal{V}}\in \mathbb{N}$ such that for any $k\in \mathbb{N}$, the open cover $\mathcal{V}\vee f^{-1}(\mathcal{V})\vee \cdots \vee f^{-k}(\mathcal{V})$ has a subcover ${\mathcal{V}}_k$ such that any $V\in {\mathcal{V}}_k$ has a periodic point of period $k+N_{\mathcal{V}}$.

Proof

Since X is a compact uniform space, using Lemma 4.2, there exists an entourage E such that for any $x\in X$, $E[x]\subset V$ for some $V\in \mathcal{V}$. Let $A=\{x_1,x_2,\cdots ,x_n\}$ be a $(k,E)-$separated subset of X with maximum cardinality. Then $\{\bigcap _{i=0}^k{f}^{-i}(E[f^i(x_j)]),j\in \{1,2,\cdots ,n\}\}$ is an open cover of X (If any $x\in X$ does not belong to $\bigcap _{i=0}^k{f}^{-i}(E[f^i(x_j)])$ for any $j\in \{1,2,\cdots ,n\}$ then $A^{\prime }=\{x_1,x_2,\cdots ,x_k,x\}$ is a $(k,E)-$separated subset of X).

Consider the open cover $\mathcal{U}=\{E[x]:x\in X\}$ of X. By the periodic open cover specification property, there exists an $N_E\in \mathbb{N}$ such that any $N_E-$spaced specification is ${\mathcal{U}}^{\prime \prime }$ traced by some periodic point of period $k+N_E$ where ${\mathcal{U}}^{\prime \prime }=\{E^{\prime \prime }[x]:x\in X\}$ and $E^{\prime \prime }$ is an entourage such that $E^{\prime \prime }\circ E^{\prime \prime }\subset E^{\prime }$, $E^{\prime }\circ E^{\prime }\subset E$. For any $j\in \{1,2,\cdots ,n\}$, consider the specification $\left(x_j,,,f,(x_j),,,\cdots ,,,f^k,(x_j)\right)$. Then there exists a $y_j\in X$ such that for any $i\in \{0,1,\cdots ,k\}$, $(f^i(x_j),f^i(y_j))\in U^{\prime \prime }\times U^{\prime \prime }$ for some $U^{\prime \prime }\in {\mathcal{U}}^{\prime \prime }$.

Take a pair of different specifications $\left(x_{j^{\prime }},,,f,(x_{j^{\prime }}),,,,\cdots ,,,,f^k,(x_{j^{\prime }})\right)$, $\left(x_{j^{\prime \prime }},,,f,(x_{j^{\prime \prime }}),,,,\cdots ,,,,f^k,(x_{j^{\prime \prime }})\right)$ such that $x_{j^{\prime }},x_{j^{\prime \prime }}\in A$. Suppose these are ${\mathcal{U}}^{\prime \prime }-$traced by $y_{j^{\prime }},y_{j^{\prime \prime }}$ respectively where $y_{j^{\prime }}$ and $y_{j^{\prime \prime }}$ are periodic points of period $k+N_E$. Then for any $i\in \{1,2,\cdots ,k\}$, $(f^i(x_{j^{\prime }}),f^i(y_{j^{\prime }}))\in E^{\prime \prime }[z^{\prime }]\times E^{\prime \prime }[z^{\prime }]$ for some $E^{\prime \prime }[z^{\prime }]\in {\mathcal{U}}^{\prime \prime }$ and $(f^i(x_{j^{\prime \prime }}),f^i(y_{j^{\prime \prime }}))\in E^{\prime \prime }[z^{\prime \prime }]\times E^{\prime \prime }[z^{\prime \prime }]$ for some $E^{\prime \prime }[z^{\prime \prime }]\in {\mathcal{U}}^{\prime \prime }$. Therefore, we have$$(f^i(x_{j^{\prime }}),z^{\prime })\in E^{\prime \prime },(f^i(y_{j^{\prime }}),z^{\prime })\in E^{\prime \prime },(f^i(x_{j^{\prime \prime }}),z^{\prime \prime })\in E^{\prime \prime }\text{and}(f^i(y_{j^{\prime \prime }}),z^{\prime \prime })\in E^{\prime \prime }.$$$$\Rightarrow (f^i(x_{j^{\prime }}),f^i(y_{j^{\prime }}))\in E^{\prime }\text{and}(f^i(x_{j^{\prime \prime }}),f^i(y_{j^{\prime \prime }}))\in E^{\prime }.$$Hence, if $y_{j^{\prime }}=y_{j^{\prime \prime }}$ then for every $i\in \{0,1,\cdots ,k\}$, $(f^i(x_{j^{\prime }}),f^i(x_{j^{\prime \prime }}))\in E$ contradicting that A is $(k,E)-$separated. Then, if $j^{\prime }\ne j^{\prime \prime }$ then the specifications $\left(x_{j^{\prime }},,,f,(x_{j^{\prime }}),,,\cdots ,,,f^k,(x_{j^{\prime }})\right)$ and $\left(x_{j^{\prime \prime }},,,f,(x_{j^{\prime \prime }}),,,\cdots ,,,f^k,(x_{j^{\prime \prime }})\right)$ are ${\mathcal{U}}^{\prime \prime }-$traced by two different points.

For any $x_j\in A$ the specification $\left(x_j,,,f,(x_j),,,\cdots ,,,f^k,(x_j)\right)$ is $U^{\prime \prime }$ traced by $y_j$ implying that $(f^i(x_j),f^i(y_j))\in E^{\prime \prime }[x_j^{\prime }]\times E^{\prime \prime }[x_j^{\prime }]$ for some $x_j^{\prime }\in X$. So, $(f^i(x_j),f^i(x))\in E^{\prime \prime }\text{and}(f^i(y_j),f^i(x))\in E^{\prime \prime }$ and we get that $(f^i(x_j),f^i(y_j))\in E^{\prime }$. Therefore, $f^i(y_j)\in E^{\prime }[f^i(x_j)]\subset E[f^i(x_j)]$. Therefore, $y_j\in f^{-i}(E[f^i(x_j)])$ for every $i\in \{0,1,\cdots ,k\}$ giving $y_j\in \bigcap _{i=0}^k{f}^{-i}(E[f^i(x_j)])$.

Now, for every $i\in \{0,1,\cdots ,k\}$ and every $j\in \{1,2,\cdots ,n\}$, $E[f^i(x_j)]\subset V_{\mathit{ij}}$ for some $V_{\mathit{ij}}\in \mathcal{V}$ which implies $\bigcap _{i=0}^k{f}^{-i}(E[f^i(x_j)])\subset \bigcap _{i=0}^k{f}^{-i}(V_{\mathit{ij}})$ and as $y_j\in \bigcap _{i=0}^k{f}^{-i}(E[f^i(x_j)]$, we get that $y_j\in \bigcap _{i=0}^k{f}^{-i}(V_{\mathit{ij}})$. Since, $\{\bigcap _{i=0}^k{f}^{-i}(E[f^i(x_j)]):j\in \{1,2,\cdots ,n\}\}$ is an open cover of X, therefore, ${\mathcal{V}}_k=\{\bigcap _{i=0}^k{f}^{-i}(V_{\mathit{ij}}):j\in \{1,2,\cdots ,n\}\}$ is also an open cover of X and every $V^{\prime }\in \{\bigcap _{i=0}^k{f}^{-i}(V_{\mathit{ij}}):j\in \{1,2,\cdots ,n\}\}$ contains $y_j$ for some $j\in \{1,2,\cdots ,n\}$, a periodic point of period $k+N_E$.

Hence, there exists an $N_{\mathcal{V}}=N_E\in \mathbb{N}$ such that for any $k\in \mathbb{N}$, the open cover $\mathcal{V}\vee f^{-1}(\mathcal{V})\vee \cdots \vee f^{-k}(\mathcal{V})$ has a subcover ${\mathcal{V}}_k$ such that any $V\in {\mathcal{V}}_k$ has a periodic point of period $k+N_{\mathcal{V}}$. $\square$

Theorem 4.2

Let (X, f) be a dynamical system with X a compact Hausdorff space. Suppose that (X, f) is positive orbit expansive and satisfies the periodic open cover specification property. Then

1. $h(f)=h(f,\mathcal{B})$ where $\mathcal{B}$ is an expansivity cover for the map f.

2. $h(f,\mathcal{B})=\underset{k\to \infty }{lim}\frac{1}{k}log(N_k(f))$.

Proof

1. Let $\mathcal{V}$ be any open cover of X. Then for any $k\in \mathbb{N}$ take the open cover ${\mathcal{V}}_k$ as defined in Theorem 4.1 and denote by ${\mathcal{V}}_k=\{V_j:j\in \{1,2,\cdots ,n\}\}$. So, every $V_j\in {\mathcal{V}}_k$ has a periodic point $y_j$ of period $k+N_{\mathcal{V}}$. By the positive orbit expansivity of (X, f), whenever $j\ne j^{\prime };j,j^{\prime }\in \{1,2,\cdots ,n\}$ there exists an $i\in \{0,1,\cdots ,k+N_{\mathcal{V}}\}$(as $y_j,y_{j^{\prime }}$ are periodic points of period $k+N_{\mathcal{V}}$) such that $(f^i(y_j),f^i(y_{j^{\prime }}))\notin B\times B$ for any $B\in \mathcal{B}$. Now, for every $j\ne j^{\prime };j,j^{\prime }\in \{1,2,\cdots ,n\}$ either $(f^i(y_j),f^i(y_{j^{\prime }}))\notin B\times B$ for some $i\in \{0,1,\cdots ,k\}$ or $(f^{k+i}(y_j),f^{k+i}(y_{j^{\prime }}))\notin B\times B$ for some $i\in \{0,1,\cdots ,N_{\mathcal{V}}\}$ or both. Let $S_k,S_{\mathcal{V}}\subset \{y_j:j\in \{1,2,\cdots ,n\}\}$ be collections of those elements of $\{y_j:j\in \{1,2,\cdots ,n\}\}$ such that for any $y_j,y_{j^{\prime }}\in S_k$ there exists an $i\in \{0,1,\cdots ,k\}$ satisfying $(f^i(y_j),f^i(y_{j^{\prime }}))\notin B\times B$ for any $B\in \mathcal{B}$ and for any $y_j^{\prime },y_{j^{\prime }}^{\prime }\in S_{\mathcal{V}}$ there exists an $i\in \{k,k+1,\cdots ,k+N_{\mathcal{V}}\}$ satisfying $(f^i(y_j^{\prime }),f^i(y_{j^{\prime }}^{\prime }))\notin B\times B$ for any $B\in \mathcal{B}$. As for two different elements $y_j,y_{j^{\prime }}\in S_k$ there exists an $i\in \{0,1,\cdots ,k\}$ satisfying $(f^i(y_j),f^i(y_{j^{\prime }}))\notin B\times B$ for any $B\in \mathcal{B}$, so $|S_k|\le N(f,\mathcal{B}\vee f^{-1}(\mathcal{B})\vee \cdots \vee f^{-k}(\mathcal{B}))$ and similarly, $|S_{\mathcal{V}}|\le N(f,\mathcal{B}\vee f^{-1}(\mathcal{B})\vee \cdots \vee f^{-N_{\mathcal{V}}}(\mathcal{B}))$. So, $N(f,\mathcal{V}\vee f^{-1}(\mathcal{V})\vee \cdots \vee f^{-k}(\mathcal{V}))\le |{\mathcal{V}}_k|\le |S_k|+|S_{\mathcal{V}}|\le N(f,\mathcal{B}\vee f^{-1}(\mathcal{B})\vee \cdots \vee f^{-k}(\mathcal{B}))+N(f,\mathcal{B}\vee f^{-1}(\mathcal{B})\vee \cdots \vee f^{-N_{\mathcal{V}}}(\mathcal{B}))$. Hence, we get $N(f,\mathcal{V}\vee f^{-1}(\mathcal{V})\vee \cdots \vee f^{-k}(\mathcal{V}))\le N(f,\mathcal{B}\vee f^{-1}(\mathcal{B})\vee \cdots \vee f^{-k}(\mathcal{B}))+N(f,\mathcal{B}\vee f^{-1}(\mathcal{B})\vee \cdots \vee f^{-N_{\mathcal{V}}}(\mathcal{B}))$ which implies that $h(f,\mathcal{V})\le h(f,\mathcal{B})$ for any open cover $\mathcal{V}$. Therefore, $h(f)=h(f,\mathcal{B}).$

2. Using Theorem 4.1, it can be easily deduced that $N(f,\mathcal{B}\vee f^{-1}(\mathcal{B})\vee \cdots \vee f^{-k}(\mathcal{B}))\le |{\mathcal{B}}_k|\le N_{k+N_{\mathcal{B}}}(f)\le N(f,\mathcal{B}\vee f^{-1}(\mathcal{B})\vee \cdots \vee f^{-(k+N_{\mathcal{B}})}(\mathcal{B})).$ Since any two points of period $k+N_{\mathcal{B}}$ cannot belong to the same open set in $\mathcal{B}\vee f^{-1}(\mathcal{V})\vee \cdots \vee f^{-k+N_{\mathcal{B}}}(\mathcal{B})$, therefore $N_{k+N_{\mathcal{B}}}\le N(f,\mathcal{B}\vee f^{-1}(\mathcal{B})\vee \cdots \vee f^{-(k+N_{\mathcal{B}})}(\mathcal{B}))$. Hence, $N(f,\mathcal{B}\vee f^{-1}(\mathcal{B})\vee \cdots \vee f^{-k}(\mathcal{B}))\le N_{k+N_{\mathcal{B}}}(f)\le N(f,\mathcal{B}\vee f^{-1}(\mathcal{B})\vee \cdots \vee f^{-(k+N_{\mathcal{B}})}(\mathcal{B}))$ implying that $\frac{1}{k}log(N(f,\mathcal{B}\vee f^{-1}(\mathcal{B})\vee \cdots \vee f^{-k}(\mathcal{B})))\le \frac{1}{k}log(N_{k+N_{\mathcal{B}}}(f))\le \frac{1}{k}log(N(f,\mathcal{B}\vee f^{-1}(\mathcal{B})\vee \cdots \vee f^{-(k+N_{\mathcal{B}})}(\mathcal{B})))$ and we get that $h(f,\mathcal{B})\le \underset{k\to \infty }{lim}\frac{1}{k}log(N_{k+N_{\mathcal{B}}}(f))\le h(f,\mathcal{B}).$ Hence, $h(f,\mathcal{B})=\underset{k\to \infty }{lim}log(N_k(f)).$$\square$

Example 4.1

Consider the map $f:S^1\to S^1$ defined by $f(e^{\iota \theta })=e^{2\iota \theta }$ where $S^1=\{e^{\iota \theta }:\theta \in \left[0,,,2,\pi \right)\}$ with the usual metric. Then (X, f) has the open cover specification property and is topologically expansive. Hence, $h(f)=\underset{k\to \infty }{lim}\frac{1}{k}log(N_k(f))$. If $f^k(e^{\iota \theta })=e^{\iota \theta }$ then $e^{2^k\iota \theta }=e^{\iota \theta }$. Hence, $2^k\iota \theta =\iota \theta +j(2\pi \iota )$ implying that $j\in \{0,1,\cdots ,2^k-2\}$. Hence, $N_k(f)=2^k-1$. Therefore, $h(f)=\underset{k\to \infty }{lim}\frac{1}{k}log(N_k(f))=\underset{k\to \infty }{lim}\frac{1}{k}log(2^k-1)=log2.$

Acknowledgements

We thank the referee for his$\backslash$her valuable comments and suggestions, which significantly contributed to the overall improvement of the paper. The first author is supported by CSIR-SRF Sr. No. 09/045(1799)/2020-EMR-I for carrying out this research work and the second author is supported by research grant under the Faculty Research Programme of the IoE scheme, University of Delhi [grant no.- IoE/2024-25/12/FRP].

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