On local aspects of sensitivity in topological dynamics

Abstract

In this paper along with the research on weakly mixing sets and transitive sets, we introduce a local aspect of sensitivity in topological dynamics and give the concept of an s-set. It is shown that a weakly mixing set is an s-set. A transitive set with the set of periodic points being dense is an s-set. In particular, a transitive set is an s-set for interval maps. Moreover, we discuss s-sets for set-valued discrete dynamical systems.

1 Introduction

A topological dynamical system (abbreviated by TDS) is a pair $(X,f)$, where X is a compact metric space with metric d and $f:X\to X$ is a continuous map. When X is finite, it is a discrete space and there is no non-trivial convergence at all. Hence, we assume that X contains infinitely many points. Let $\mathbb{N^{+}}$ denote the set of all positive integers and let $\mathbb{N}=\mathbb{N^{+}}\cup \{0\}$.

Transitivity, weak mixing, and sensitive dependence on initial conditions (see [1–4]) are global characteristics of topological dynamical systems. Let $(X,f)$ be a TDS, $(X,f)$ is (topologically) transitive if for any nonempty open subsets U and V of X there exists an $n\in\mathbb{N}$ such that $f^{n}(U)\cap V\neq\emptyset$. $(X,f)$ is (topologically) mixing if for any nonempty open subsets U and V of X, there exists an $N\in\mathbb{N}$ such that $f^{n}(U)\cap V\neq\emptyset$ for all $n\in\mathbb{N}$ with $n\geq N$. $(X,f)$ is (topologically) weakly mixing if for any nonempty open subsets $U_{1}$, $U_{2}$, $V_{1}$, and $V_{2}$ of X, there exists an $n\in\mathbb{N}$ such that $f^{n}(U_{1})\cap V_{1}\neq\emptyset$ and $f^{n}(U_{2})\cap V_{2}\neq\emptyset$. It follows from these definitions that mixing implies weak mixing, which in turn implies transitivity. A map f is said to have sensitive dependence on initial conditions if there is a constant $\delta>0$ such that for any nonempty open set U of X, there exist points $x,y\in U$ such that $d(f^{n}(x),f^{n}(y))> \delta$ for $n\in\mathbb{N^{+}}$.

In [5], Blanchard introduced overall properties and partial properties. For example, sensitive dependence on initial conditions, Devaney chaos (see [6]), weak mixing, mixing and more belong to overall properties; Li-Yorke chaos (see [7]) and positive entropy (see [1, 8]) belong to partial properties. Weak mixing is an overall property, it is stable under semi-conjugate maps and implies Li-Yorke chaos. We find that a weakly mixing system always contains a dense uncountable scrambled set (see [9]). In [10], Blanchard and Huang introduced the concepts of a weakly mixing set, derived from a result given by Xiong and Yang [11] and showed ‘partial weak mixing implies Li-Yorke chaos’ and ‘Li-Yorke chaos cannot imply partial weak mixing’.

Motivated by the idea of Blanchard and Huang’s notion of a ‘weakly mixing subset’, Oprocha and Zhang [12] extended the notion of a weakly mixing set and gave the concept of a transitive set and discussed its basic properties. In this paper we give the concept of ‘s-set’ for topological dynamical systems and investigate the relationship among transitive subsets, weakly mixing sets, and s-sets. We find that a TDS to have a weakly mixing set implies it has an s-set, and if periodic points are dense in the transitive set, then the transitive set is an s-set. In particular, a transitive set is an s-set for interval maps. The properties of transitivity, weak mixing, and sensitivity on initial conditions for a set-valued discrete dynamical system were discussed (see [13–18]). Also, we continue to discuss s-sets for set-valued discrete dynamical systems and investigate the relationship between a set-valued discrete system and an original system on an s-set. More precisely, a set-valued discrete system has an s-set, which implies that an original system has an s-set.

2 Preliminaries

A TDS $(X,f)$ is point transitive if there exists a point $x_{0}\in X$ with dense orbit i.e., $\overline{\operatorname{orb}(x_{0})}=X$. Such a point $x_{0}$ is called a transitive point of $(X,f)$. In general, transitive and point transitive are independent (see [2]). A TDS $(X,f)$ is minimal if $\overline{\operatorname{orb}(x,f)}=X$ for every $x\in X$, i.e., every point is transitive point. A point x is called minimal if the subsystem $(\overline{\operatorname{orb}(x,f)},f)$ is minimal. Denote by $P(f)$ the set of all periodic points.

A Li-Yorke or scrambled pair $(x,y)$ of $(X,f)$ is a pair of points of X such that

• (1) $\lim _{n\to\infty}\inf d(f^{n}(x),f^{n}(y))=0$, and

• (2) $\lim _{n\to\infty}\sup d(f^{n}(x),f^{n}(y))>0$.

A set $S\subseteq X$ is said to be a scrambled set if for all $x,y\in S$, $x\neq y$, the $(x,y)$ is a scrambled pair. A TDS $(X,f)$ is called Li-Yorke chaotic if it contains an uncountable scrambled set.

A map f is said to be Devaney chaotic if f satisfies the following conditions:

• (1) f is transitive,

• (2) f is periodically dense; i.e., the set of periodic points of f is dense in X, and

• (3) f is sensitive dependent on initial conditions.

Definition 2.1

[10]

Let $(X,f)$ be a TDS and A be a closed subset of X with at least two elements. A is said to be weakly mixing if for any $k\in\mathbb{N}$, any choice of nonempty open subsets $V_{1},V_{2},\ldots,V_{k}$ of A and nonempty open subsets $U_{1},U_{2},\ldots,U_{k}$ of X with $A\cap U_{i}\neq\emptyset$, $i=1,2,\ldots,k$, there exists an $m\in\mathbb{N}$ such that $f^{m}(V_{i})\cap U_{i}\neq\emptyset$ for $1\leq i\leq k$. $(X,f)$ is called partial weak mixing if X contains a weakly mixing subset.

Definition 2.2

[12]

Let $(X,f)$ be a TDS and A be a nonempty subset of X. A is called a transitive set of $(X,f)$ if, for any choice of nonempty open subset $V^{A}$ of A and nonempty open subset U of X with $A\cap U\neq\emptyset$, there exists an $n\in\mathbb{N}$ such that $f^{n}(V^{A})\cap U\neq\emptyset$.

Remark 2.1

• (1) $(X,f)$ is topologically transitive if and only if X is a transitive set of $(X,f)$.

• (2) By [12], A is a transitive set if and only if A̅ is a transitive set, where A̅ denotes the closure of A.

According to the definitions of transitive set and weakly mixing subset, we have the following results.

Result 1.:

If A is a weakly mixing set of $(X,f)$, then A is a transitive set of $(X,f)$.

Result 2.:

If $a\in X$ is a transitive point of $(X,f)$, then $\{a\}$ is a transitive set of $(X,f)$.

Result 3.:

If $A=\operatorname{orb}(x,f)$ is a periodic orbit of $(X,f)$ for some $x\in X$, then A is a transitive set of $(X,f)$.

Definition 2.3

A nonempty subset A is called an s-set of $(X,f)$ if there exists a $\delta>0$ such that for any $x\in A$ and $\varepsilon>0$, there exist a $y\in B(x,\varepsilon)\cap A$ and an $n\in\mathbb{N^{+}}$ satisfying $d(f^{n}(x),f^{n}(y))>\delta$.

Remark 2.2

The s-set is dense in itself, i.e., it contains no isolated points. X is an s-set of $(X,f)$ if and only if $(X,f)$ is sensitive dependent on initial conditions.

Let $(X,f)$ and $(Y,g)$ be two TDSs. Then $(X,f)$ is an extension of $(Y,g)$, or $(Y,g)$ is a factor of $(X,f)$ if there exists a surjective continuous map $h: X\to Y$ (called a factor map) such that $h\circ f(x)=g\circ h(x)$ for every $x\in X$. If further h is a homeomorphism, then $(X,f)$ and $(Y,g)$ are said to be topologically conjugate and the homeomorphism h is called a conjugated map.

Theorem 2.1

Let A be a weakly mixing set of $(X,f)$. Then A is an s-set of $(X,f)$.

Proof

Let A be a weakly mixing set of $(X,f)$. Then A contains at least two points. Pick up two distinct points $x_{1},x_{2}\in A$ and set $\delta=\frac{1}{4}d(x_{1},x_{2})>0$. For any $x\in A$ and $\varepsilon>0$, we see that $B(x,\varepsilon)$ is a nonempty open subset of X.

We consider an open subset $B(x,\varepsilon)\cap A$ of A and two open subsets $B(x_{1},\delta)$, $B(x_{2},\delta)$ of X. Since A is a weakly mixing set and $B(x_{1},\delta)\cap A\neq\emptyset$, $B(x_{2},\delta)\cap A\neq\emptyset$, there exists an $n\in\mathbb{N}$ such that $f^{n}(B(x,\varepsilon)\cap A)\cap B(x_{1},\delta)\neq\emptyset$ and $f^{n}(B(x,\varepsilon)\cap A)\cap B(x_{2},\delta)\neq\emptyset$. Furthermore, there exist $y_{1},y_{2}\in B(x,\varepsilon)\cap A$ such that $f^{n}(y_{1})\in B(x_{1},\delta)$ and $f^{n}(y_{2})\in B(x_{2},\delta)$. Moreover,

$$\begin{aligned} d\bigl(f^{n}(y_{1}),f^{n}(y_{2})\bigr) \geq d(x_{1},x_{2})-d\bigl(x_{1},f^{n}(y_{1}) \bigr)-d\bigl(x_{2},f^{n}(y_{2})\bigr)>4\delta- \delta-\delta=2\delta. \end{aligned}$$

Therefore, either $d(f^{n}(x),f^{n}(y_{1}))>\delta$ or $d(f^{n}(x),f^{n}(y_{2}))>\delta$. This shows that A is an s-set of $(X,f)$. □

3 Characterizing s-sets

In this section, we discuss the properties of s-sets of $(X,f)$. For a TDS $(X,f)$ and two nonempty subsets $U,V\subseteq X$, we use the following notation:

$$\begin{aligned} N(U,V)=\bigl\{ {n\in\mathbb{N}}: f^{n}(U)\cap V\neq\emptyset\bigr\} . \end{aligned}$$

Lemma 3.1

Let A be an infinite subset of X and $P(f)$ be dense in A. Then $P(f)\cap A$ is infinite.

Proof

Suppose that $P(f)\cap A$ is finite and let $\operatorname{card}(P(f)\cap A)=n$, where card represents the cardinality of a set. Since A is an infinite subset of X, there exists a pairwise disjoint open set $V_{i}^{A}$ of A for $i=1,2,\ldots,n+1$, i.e., $V_{i}^{A}\cap V_{j}^{A}=\emptyset$ for $i,j\in\{1,2,\ldots ,n+1\}$ and $i\neq j$. Moreover, $P(f)\cap A$ is dense in A, which implies $\operatorname{card}(P(f)\cap A)\geq n+1$. This is a contradiction. Therefore, $P(f)\cap A$ is infinite. □

Theorem 3.1

Let $(X,f)$ be a TDS and A be an infinite subset of X. If A is a transitive set of $(X,f)$ and $P(f)$ is dense in A, then A is an s-set of $(X,f)$.

Proof

We first prove that there exists $\delta_{0}>0$ such that, for any $x\in A$, there exists $q\in P(f)\cap A$ satisfying

$$\begin{aligned} d\bigl(\operatorname{orb}(q),x\bigr)\geq\frac{\delta_{0}}{2}. \end{aligned}$$

Indeed, by Lemma 3.1, $P(f)\cap A$ is an infinite set. Hence, we pick two different points $q_{1},q_{2}\in P(f)\cap A$ such that $\operatorname{orb}(q_{1})\cap \operatorname{orb}(q_{2})=\emptyset$. Let

$$\begin{aligned} \delta_{0}=d\bigl(\operatorname{orb}(q_{1}),\operatorname{orb}(q_{2})\bigr)>0. \end{aligned}$$

(3.1)

Then, for any $x\in A$, we have

$$\begin{aligned} d\bigl(\operatorname{orb}(q_{1}),x\bigr)>\frac{\delta_{0}}{2} \quad\mbox{or}\quad d \bigl(\operatorname{orb}(q_{2}),x\bigr)>\frac{\delta_{0}}{2}. \end{aligned}$$

(3.2)

If (3.2) is false, then

$$\begin{aligned} d\bigl(\operatorname{orb}(q_{1}),x\bigr)\leq\frac{\delta_{0}}{2}\quad \mbox{and}\quad d \bigl(\operatorname{orb}(q_{2}),x\bigr)\leq\frac{\delta_{0}}{2}. \end{aligned}$$

Hence, by the triangle inequality, we have

$$\begin{aligned} d\bigl(\operatorname{orb}(q_{1}),\operatorname{orb}(q_{2})\bigr)< \delta_{0}. \end{aligned}$$

This is a contradiction by (3.1).

Take $\delta=\frac{\delta_{0}}{8}$. For any $x\in A$ and $\varepsilon>0$, without loss of generality, let $\varepsilon<\delta$. Since $P(f)$ is dense in A, we have $P(A)\cap(B(x,\varepsilon)\cap A)\neq\emptyset$. Furthermore, we can take $p\in B(x,\varepsilon)\cap A$ and let $f^{n}(p)=p$. Since $x\in A$, there exists $q\in P(f)\cap A$ such that

$$\begin{aligned} d\bigl(\operatorname{orb}(q),x\bigr)\geq4\delta. \end{aligned}$$

Let $U=\bigcap _{i=1}^{n}f^{-i}(B(f^{i}(q),\delta))$. Since $q\in U$, we have $q\in U\cap A$, which implies $U\cap A\neq\emptyset$. Moreover, A is a transitive set of $(X,f)$, thus there exists a $k\in\mathbb{N^{+}}$ such that $B(x,\varepsilon)\cap A\cap f^{-k}(U)\neq\emptyset$. Take $y\in B(x,\varepsilon)\cap A\cap f^{-k}(U)$. Then $f^{k}(y)\in U$. Let $j=[\frac{k}{n}+1]$. Then $1\leq nj-k\leq n$. Furthermore, we have

$$\begin{aligned} f^{nj}(y)\in f^{nj-k}\bigl(f^{k}(y)\bigr)\in f^{nj-k}(U)\subseteq B\bigl(f^{nj-k}(q),\delta\bigr). \end{aligned}$$

Since $f^{n}(p)=p$, we have $f^{nj}(p)=p$. Hence, by the triangle inequality,

$$\begin{aligned} d\bigl(f^{nj}(p),f^{nj}(y)\bigr)=d\bigl(p,f^{nj}(y) \bigr)\geq d\bigl(x,f^{nj-k}(q)\bigr)-d\bigl(f^{nj-k}(q),f^{nj}(y) \bigr)-d(p,x). \end{aligned}$$

As $p\in B(x,\delta)\cap A$ and $f^{nj}(y)\in B(f^{nj-k}(q),\delta)$, so

$$\begin{aligned} d\bigl(f^{nj}(p),f^{nj}(y)\bigr)>4\delta-\delta-\delta=2 \delta. \end{aligned}$$

Again, by the triangle inequality, we have

$$\begin{aligned} d\bigl(f^{nj}(x),f^{nj}(y)\bigr)>\delta \quad\mbox{or}\quad d \bigl(f^{nj}(x),f^{nj}(p)\bigr)>\delta, \end{aligned}$$

for some $p\in B(x,\varepsilon)\cap A$ and some $y\in B(x,\varepsilon)\cap A$. Therefore, A is an s-set of $(X,f)$.

By [19], if X is a non-degenerate compact interval, $f:X\to X$ is a continuous map and f is transitive, then $P(f)$ is dense in X. We prove that if X is a non-degenerate compact interval, A is a non-degenerate closed interval, and A is a transitive set of $(X,f)$, then $P(f)$ is dense in A. □

Lemma 3.2

[19]

Suppose that I is a non-degenerate interval and $f:I\to I$ is a continuous map. If $J\subseteq I$ is an interval which contains no periodic points of f and z, $f^{m}(z)$ and $f^{n}(z)\in J$ with $0< m< n$, then either $z< f^{m}(z)< f^{n}(z)$ or $z>f^{m}(z)>f^{n}(z)$.

Theorem 3.2

Let I be a non-degenerate interval and $f:I\to I$ be a continuous map. If $J\subseteq I$ is a non-degenerate closed interval and J is a transitive set of $(I,f)$, then $P(f)$ is dense in J.

Proof

Suppose that $P(f)$ is not dense in J. Then there exists a nonempty open set $J_{1}$ of J containing no periodic points, i.e., $P(f)\cap J_{1}=\emptyset$. Without loss of generality, let $J_{1}$ be an open interval of I and $J_{1}\subseteq J$. Take an $x\in J_{1}$ which is not an endpoint of $J_{1}$, an open neighborhood $U\subsetneq J_{1}$ of x and an open interval $E\subseteq J_{1}\setminus U$.

We consider open neighborhood U of $J_{1}$ and open interval E with $J_{1}\cap E\neq\emptyset$. Since J is a transitive set of $(I,f)$, there exists an $m\in\mathbb{N}$ such that $f^{m}(U)\cap E\neq\emptyset$. Furthermore, there exists a $y\in U\subseteq J_{1}$ such that $f^{m}(y)\in E\subseteq J_{1}$. Moreover, $P(f)\cap J_{1}=\emptyset$, it means that $y\neq f^{m}(y)$. Since I is a Hausdorff space and f is continuous, there exists an open neighborhood V of y such that $f^{m}(V)\cap V=\emptyset$. Hence, we can take an open interval $J_{2}$ of I such that $J_{2}\subseteq J_{1}$ and $y\in J_{2}\subseteq V$, thus $f^{m}(J_{2})\cap J_{2}=\emptyset$. Since $J_{2}$ is an open interval of I and $J_{2}\subseteq J_{1}$ and J is a transitive set, there exist $n>m$ and $z\in U$ such that $f^{n}(z)\in J_{2}$. Furthermore, we have $0< m< n$ and $z, f^{n}(z)\in J_{2}$ while $f^{m}(z)\notin J_{2}$. This is a contradiction by Lemma 3.2. Therefore, $P(f)$ is dense in J. □

Example 3.1

We have the tent map (see Figures 1 and 2)

$$\begin{aligned} f(x)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} 2x, &\mbox{if } 0\leq x\leq\frac{1}{2},\\ 2(1-x),&\mbox{if } \frac{1}{2}\leq x\leq1, \end{array}\displaystyle \right . \end{aligned}$$

which is called Devaney chaos on $I=[0,1]$ by [6]. We will prove that $[\frac{1}{4},\frac{3}{4}]$ is a transitive set of $(I,f)$.

Figure 1

The tent map f .

Figure 2

The 2nd iterate $\pmb{f^{2}}$ of the tent map f .

Let $S(f^{k})$ denote the set of extreme value points of $f^{k}$ for every $k\in\mathbb{N^{+}}$. Then $S(f^{k})=\{\frac{1}{2^{k}},\frac{2}{2^{k}},\ldots,\frac{2^{k}-1}{2^{k}}\}$. Since $S(f)=\{\frac{1}{2}\}$, $f(\frac{1}{2})=1$, $f(0)=0$, and $f(1)=0$, we have

$$\begin{aligned} f^{k}(x)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} 1,&\mbox{if } x=\frac{1}{2^{k}}, \frac{3}{2^{k}},\ldots, \frac{2^{k}-1}{2^{k}},\\ 0,&\mbox{if } x=0, \frac{2}{2^{k}}, \frac{4}{2^{k}},\ldots, \frac{2^{k}-2}{2^{k}}, 1. \end{array}\displaystyle \right . \end{aligned}$$

Let $I_{k}^{j}=[\frac{j}{2^{k}},\frac{j+1}{2^{k}}]$ for $0\leq j\leq2^{k}-1$. Then $f^{k}(I_{k}^{j})=[0,1]$. For any nonempty open set U of $[\frac{1}{4},\frac{3}{4}]$, without loss of generality, we take $U=(x_{0}-\varepsilon,x_{0}+\varepsilon)$ for a given $\varepsilon>0$ and $x_{0}\in \operatorname{int}[\frac{1}{4},\frac{3}{4}]$, where $\operatorname{int}[\frac{1}{4},\frac{3}{4}]$ denotes the interior of $[\frac{1}{4},\frac{3}{4}]$. When $l\in\mathbb{N}$ and $l>\log_{2}\frac{1}{\varepsilon}$, there exists j, $j\in\mathbb{N}$ and $0\le j\le2^{l}-1$, such that $I_{l}^{j}\subseteq U$. Furthermore, we have $f^{l}(U)=[0,1]$. Thus, for any nonempty open set U of $[\frac{1}{4},\frac{3}{4}]$ and nonempty open set V of $[0,1]$ with $V\cap[\frac{1}{4},\frac{3}{4}]\neq\emptyset$, there exists a $k\in\mathbb{N}$ such that $f^{k}(U)\cap V\neq\emptyset$. This shows that $[\frac{1}{4},\frac{3}{4}]$ is a transitive set of $(I,f)$. Since $P(f)$ is dense in I, $P(f)$ is also dense in $[\frac{1}{4},\frac{3}{4}]$. By Theorem 3.2, $[\frac{1}{4},\frac{3}{4}]$ is an s-set of $(I,f)$.

4 s-sets for set-valued discrete dynamical systems

The distance from a point x to a nonempty set A in X is defined by

$$\begin{aligned} d(x,A)=\inf_{a\in A}d(x,a). \end{aligned}$$

Let $\kappa(X)$ be the family of all nonempty compact subsets of X. The Hausdorff metric on $\kappa(X)$ is defined by

$$\begin{aligned} d_{H}(A,B)=\max\Bigl\{ \sup _{a\in A}d(a,B), \sup _{b\in B}d(b,A) \Bigr\} \quad\mbox{for every } A,B\in\kappa(X). \end{aligned}$$

It follows from Michael [20] and Engelking [21] that $\kappa(X)$ is a compact metric space. The Vietoris topology $\tau_{\upsilon}$ on $\kappa(X)$ is generated by the base

$$\begin{aligned} \upsilon(U_{1},U_{2},\ldots,U_{n})=\Biggl\{ F\in \kappa(X): F\subseteq\bigcup _{i=1}^{n} U_{i} \mbox{ and } F\cap U_{i}\neq\emptyset\mbox{ for all } i\leq n\Biggr\} , \end{aligned}$$

where $U_{1},U_{2},\ldots,U_{n}$ are open subsets of X. Let f̄ be the induced set-valued map defined by

$$\begin{aligned} \bar{f}: \kappa(X)\to\kappa(X), \quad\bar{f}(F)=f(F), \quad\mbox{for every }F\in \kappa(X). \end{aligned}$$

Then f̄ is well defined. $(\kappa(X),\bar{f})$ is called a set-valued discrete dynamical system.

Let X be a $T_{1}$ space, that is, a single point set that is closed. Then $\kappa(A)=\{F\in\kappa(X): F\subseteq A\}$ is a closed subset of $\kappa(X)$ for any nonempty closed subset A of X (see [20]).

Theorem 4.1

Let A be a nonempty closed subset of X. If $P(f)$ is dense in A, then $P(\bar{f})$ is dense in $\kappa(A)$.

Proof

Let ${\mathcal{V}}^{\kappa(A)}$ be a nonempty open subset of $\kappa(A)$. Then there exists an open set ${\mathcal{V}}$ of $\kappa (X)$ such that ${\mathcal{V}}^{\kappa(A)}={\mathcal{V}}\cap\kappa(A)$. Without loss of generality, let ${\mathcal{V}}=\nu(V_{1},V_{2},\ldots,V_{m})$. Take $F\in{\mathcal{V}}\cap\kappa(A)$. Then we have $F\subseteq A$, $F\subseteq\bigcup _{i=1}^{m} V_{i}$ and $F\cap V_{i}\neq\emptyset$ for each $i=1,2,\ldots,m$. Hence, $V_{i}\cap A\neq\emptyset$ for each $i=1,2,\ldots,m$. Since $P(f)$ is dense in A, it follows that $P(f)\cap(V_{i}\cap A)\neq\emptyset$ for each $i=1,2,\ldots,m$. Furthermore, there exist $y_{i}\in P(f)\cap(V_{i}\cap A)$ and $n_{i}\in\mathbb{N^{+}}$ such that $f^{n_{i}}(y_{i})=y_{i}$ for each $i=1,2,\ldots,m$. Let $G=\{y_{1},y_{2},\ldots,y_{m}\}$. Then $G\in {\mathcal{V}}$ and $G\in\kappa(A)$, which implies $G\in {\mathcal{V}}^{\kappa(A)}$. Moreover, $f^{n_{1}n_{2}\cdots n_{m}}(y_{i})=y_{i}$ for each $i=1,2,\ldots,m$. Therefore, $(\bar{f})^{n_{1}n_{2}\cdots n_{m}}(G)=f^{n_{1}n_{2}\cdots n_{m}}(G)=G$, it means that $P(\bar{f})\cap {\mathcal{V}}^{\kappa(A)}\neq\emptyset$. This shows that $P(\bar{f})$ is dense in $\kappa(A)$. □

Theorem 4.2

Let A be a nonempty closed subset of X. If $\kappa(A)$ is a sensitive set of $(\kappa(X),\bar{f})$, then A is an s-set of $(X,f)$.

Proof

Since $\kappa(A)$ is an s-set of $(\kappa(X),\bar{f})$, there exists a constant $\delta>0$ such that $K\in\kappa(A)$ and every $\varepsilon>0$ there exist $G\in B(K,\varepsilon)\cap\kappa(A)$ and $n\in\mathbb{N^{+}}$ such that $d_{H}((\bar{f})^{n}(K),(\bar{f})^{n}(G))>\delta$.

Let $x\in A$ and $\varepsilon>0$. Take $K=\{x\}\in\kappa(A)$. Then there exist $G\in B(\{x\},\varepsilon)\cap\kappa(A)$ and $n\in\mathbb{N^{+}}$ such that

$$\begin{aligned} d_{H}\bigl((\bar{f})^{n}\bigl(\{x\}\bigr),( \bar{f})^{n}(G)\bigr)=d_{H}\bigl(f^{n}\bigl(\{x\} \bigr),f^{n}(G)\bigr)\geq\delta. \end{aligned}$$

Since $d_{H}(f^{n}(\{x\}),f^{n}(G))=\sup _{y\in G}d(f^{n}(x),f^{n}(y))$, G is a compact subset of X and $f: X\to X$ is a continuous map, there exists $y_{0}\in G$ such that

$$\begin{aligned} d_{H}\bigl(f^{n}\bigl(\{x\}\bigr),f^{n}(G) \bigr)=d\bigl(f^{n}(x),f^{n}(y_{0})\bigr)>\delta. \end{aligned}$$

Moreover, $G\in B(\{x\},\varepsilon)\cap\kappa(A)$ implies $G\subseteq B(x,\varepsilon)$ and $G\subseteq A$, consequently, $y_{0}\in B(x,\varepsilon)\cap A$. This shows that A is an s-set of $(X,f)$. □