The Ergodicity and Sensitivity of Nonautonomous Discrete Dynamical Systems

Abstract

Let $(E,h_{1,\infty })$ be a nonautonomous discrete dynamical system (briefly, N.D.D.S.) that is defined by a sequence ${\left(h_j\right)}_{j=1}^{\infty }$ of continuous maps $h_j:E\to E$ over a nontrivial metric space $(E,d)$. This paper defines and discusses some forms of ergodicity and sensitivity for the system $(E,h_{1,\infty })$ by upper density, lower density, density, and a sequence of positive integers. Under some conditions, if the rate of convergence at which ${\left(h_j\right)}_{j=1}^{\infty }$ converges to the limit map h is “fast enough” with respect to a sequence of positive integers with a density of one, it is shown that several sensitivity properties for the N.D.D.S. $(E,h_{1,\infty })$ are the same as those properties of the system $(E,h)$. Some sufficient conditions for the N.D.D.S. $(E,h_{1,\infty })$ to have stronger sensitivity properties are also presented. The conditions in our results are less restrictive than those in some existing works, and the conclusions of all the theorems in this paper improve upon those of previous studies. Thus, these results are extensions of the existing ones.

1. Introduction

After Li and Yorke [1] first gave the notion of “chaos”, many scholars began to study the Li–Yorke chaotic autonomous discrete systems (briefly, A.D.S.) [2,3,4,5,6,7,8]. Schweizer and Smítal [5] defined distributional chaos. The distributionally chaotic A.D.S.s have been studied by many researchers (for example, [2,3,9,10]). J. C. Xiong et al. [2] described chaos by using Furstenberg families and showed that chaos in the sense of Li–Yorke and a few versions of distributional chaos could be considered chaos in the Furstenberg family sense. In [11], H. Y. Wang et al. presented some notions of sensitivity by using Furstenberg families and discussed a few of their related properties. By exploring these notions, they also obtained some meaningful results. In [3], Tan and Xiong introduced the concept of $(\mathcal{F}1,\mathcal{F}2)$-chaos, which involves the use of the Furstenberg family couple $\mathcal{F}1$ and $\mathcal{F}2$. They also provided sufficient conditions for a system to be $(\mathcal{F}1,\mathcal{F}2)$-chaotic and presented an interesting example.

N.D.D.S.s were defined in [12,13]. We know that N.D.D.S.s also appear connected to some nonautonomous difference equations. N.D.D.S.s chaos has been extensively explored [12,13,14,15]. For N.D.D.S.s, exploring the conditions under which a chaotic property under limited operations is preserved is a significant problem [14,16,17,18,19]. For example, H. Román-Flores [18] showed that if a sequence of continuous maps ${\left(h_m\right)}_{m=1}^{\infty }$ over a metric space $(E,d)$ converges uniformly to a map h of E, and $h_m$ is topologically transitive for any $m\in 1,2,\cdots$, it does not necessarily mean that the limit map h is also topologically transitive. He also gave several sufficient conditions where the limit map h was topologically transitive. A. Fedeli et al. [17] explored the dynamical properties of the limit map of a sequence of topological transitivity maps over a given compact metric space and gave some conditions where the limit map was topologically transitive. In [15], J. S. Cánovas explored the dynamical properties of the limit map of mapping sequences with the form $h_m\circ \cdots \circ h_1$ (where $h_k$ is a continuous map over space $E=[0,1]$ for every positive integer k). Then, The question is whether the simplicity (or chaoticity) of h implies the simplicity (or chaoticity) of $h_{1,\infty }$, where $h_{1,\infty }={\left(h_k\right)}_{k=1}^{\infty }$ denotes a sequence of continuous maps over the space E converging uniformly to a continuous map h of the same space E. In [14], the author proved that the full Lebesgue measure of a distributionally scrambled set of the N.D.D.S. cannot deduce that the limit map is distributionally chaotic. There is a N.D.D.S. $h_{1,\infty }$, such that its distributionally scrambled set is arbitrarily small; it satisfies that the sequence ${\left(h_k\right)}_{k=1}^{\infty }$ can converge uniformly to a limit map and the limit map is distributionally chaotic almost everywhere. In [20], R. Vasisht et al. considered some stronger forms of transitivity for a N.D.D.S. $(E,h_{1,\infty })$ that is defined by a sequence ${\left(h_k\right)}_{k=1}^{\infty }$ of continuous maps that converge uniformly to a map h. They introduced the notions of thick sensitivity, ergodic sensitivity, and multi-sensitivity for N.D.D.S.s. They showed that under certain conditions, if the sequence $\left(h_k\right){k=1}^{\infty }$ converges to the limit map h at a “sufficiently fast” rate, then the sensitivity and transitivity properties of the N.D.D.S. $(E,h_1,\infty )$ and the limit system $(E,h)$ will coincide. Inspired by [3,11,21], we further studied the transitivity properties and the sensitivity properties for an N.D.D.S. $(E,h_{1,\infty })$ and extended some responding results by R. Vasisht et al. in [20]. In particular, under the conditions that $h_k$ is semi-open, $h_k\circ h=h\circ h_k$ for every positive integer k, and ${\displaystyle \sum _{k=1}^{\infty }}D(h_k,h)$ exists (i.e., ${\displaystyle \sum _{k=1}^{\infty }}D(h_k,h)<+\infty$, where $D(h_k,h)=\underset{x\in E}{lim}d(h_k\left(x\right),h\left(x\right))$), ref. [21] obtained that $(E,h_{1,\infty })$ is $\mathcal{G}$-transitive (resp. $\mathcal{G}$-mixing or $\mathcal{G}$-sensitive or $({\mathcal{G}}_1,{\mathcal{G}}_2)$-sensitive or $\mathcal{G}$-collectively sensitive or $\mathcal{G}$-synchronous sensitive or $\mathcal{G}$-multi-sensitive) if and only if $(E,h)$, where $\mathcal{G}$ is a family.

In this paper, by upper density, lower density, density, and a sequence of positive integers, some stronger forms of ergodicity and sensitivity for N.D.D.S.s are considered. Moreover, topological ergodicity, topologically strong-ergodicity, upper density one sensitivity, density one sensitivity, ergodic sensitivity, ergodic multi-sensitivity, ergodically collective sensitivity, and ergodically synchronous sensitivity for the system $(E,h_{1,\infty })$ are discussed. Under the conditions that $h_k$ is semi-open, $h_k\circ h=h\circ h_k$ for each $k\in \{1,2,\cdots \}$, and that there is a subset $A\subset \{1,2,\cdots \}$, such that $\mu \left(A\right)=1$ and

$$\underset{k\in A,k\to \infty }{lim}\sum _{i=1}^{k}D(h_i,h)$$

exists, some sufficient conditions or necessary and sufficient conditions about chaoticity between $(E,h)$ and $(E,h_{1,\infty })$ are established. Since the conditions of the above results relax and extend the conditions of some responding results in [20], and conclusions of all theorems in this paper enhance those in responding theorems in [20], the above results extend and improve the responding ones in [20]. Moreover, we obtain several new results.

Section 2 recalls and introduces some notations and basic notions. The main theorems are obtained and shown in Section 3.

2. Preliminaries

Throughout this paper, $(E,h_{1,\infty })$ is always assumed to be a N.D.D.S., which is defined by sequences ${\left(h_k\right)}_{k=1}^{\infty }$ of continuous maps $h_k:E\to E$ over a nontrivial metric space $(E,d)$. If $h_k=h$ for each integer $k\ge 1$, then the pair $(E,h)$ is said to be a ‘classical’ autonomous discrete dynamical system (briefly, A.D.D.S.).

Let $I=\{k_i\mid i\in {\mathbb{Z}}^{+},k_1<k_2<\cdots \}\subset {\mathbb{Z}}^{+}$,

$$\overline{\mu }\left(I\right)=\underset{n\to \infty }{lim\; sup}\frac{|I\cap \{1,2,\cdots ,n\left\}\right|}{n}$$

and

$$\underline{\mu }\left(I\right)=\underset{n\to \infty }{lim\; inf}\frac{|I\cap \{1,2,\cdots ,n\left\}\right|}{n},$$

then $\overline{\mu }\left(I\right)$ is said to be the upper density of $I\subset \{0,1,\cdots \}$, and $\underline{\mu }\left(I\right)$ is said to be the lower density of $I\subset \{0,1,\cdots \}$ ([22]). If $\overline{\mu }\left(I\right)=\underline{\mu }\left(I\right)$, it is said to be a density of I and denoted by $\mu \left(I\right)$. It is easily seen that

$$\overline{\mu }\left(I\right)=\underset{j\to \infty }{lim\; sup}\frac{j}{k_j} \mathrm{and} \underline{\mu }\left(I\right)=\underset{j\to \infty }{lim\; inf}\frac{j}{k_j}.$$

For any $b\in [0,1]$, let

$$\overline{\mathcal{M}}\left(b\right)=\{F\in \mathcal{B}:\overline{\mu }\left(F\right)\ge b\}.$$

It is clear that $\overline{\mathcal{M}}\left(0\right)=\mathcal{B}$, where $\mathcal{B}$ denotes the family of all infinite subsets of the nonnegative integer set.

For a N.D.D.S. $(E,h_{1,\infty })$, any $F,G\subset E$ and any $\eta >0$, let

$$N_{h_{1,\infty }}(F,G)=\{k\in {\mathbb{Z}}^{+}:h_1^k\left(F\right)\cap G\ne Ø\}=\{k\in {\mathbb{Z}}^{+}:F\cap h_1^{-k}\left(G\right)\ne Ø\}$$

and $N_{h_{1,\infty }}(F,\eta )=\{k\in {\mathbb{Z}}^{+}:$ there are $f,g\in F$ with $d(h_1^k\left(f\right),h_1^k\left(g\right))\ge \eta \}$, where $h_1^k=h_k\circ \cdots \circ h_1$, $h_1^{-k}\left(G\right)=h_1^{-1}\circ h_2^{-1}\circ \cdots \circ h_k^{-1}\left(G\right)$.

Definition 1.

An NDDS $(E,h_{1,\infty })$ is called topologically ergodic if $\overline{\mu }\left(N_{h_{1,\infty }}(F,G)\right)>0$ for any nonempty open sets $F,G\subset E$. An NDDS $(E,h_{1,\infty })$ is called topologically strong-ergodic if $\overline{\mu }\left(N_{h_{1,\infty }}(F,G)\right)=1$ for any nonempty open sets $F,G\subset E$.

Definition 2.

The system $(E,h_{1,\infty })$ is said to be an ergodically collectively sensitive system with the collective sensitivity constant η$(\eta >0)$ if for every integer $n>0$, any $a_1,a_2,\cdots ,a_n\in E$ with $a_i\ne a_j$ for any two integers $i,j\in \{1,2,\cdots ,n\}$ with $i\ne j$ and every $\epsilon >0$, there are $b_1,b_2,\cdots ,b_n\in E$ with $b_k\ne b_l$ for any $k,l\in \{1,2,\cdots ,n\}$ with $k\ne l$ satisfying the following:

(1)

$d(a_j,b_j)<\epsilon$ for all $1\le j\le n$;

(2)

there exists a $j_0\in \mathbb{N}$ with $1\le j_0\le n$ and $A\subset \{1,2,\cdots \}$ with $\overline{\mu }\left(A\right)>0$, such that for any $k\in A$, one has that $d(h_1^k\left(a_j\right),h_1^k\left(b_{j_0}\right))>\eta$ for every $j\in \{1,2,\cdots ,n\}$ or $d(h_1^k\left(a_{j_0}\right),h_1^k\left(b_j\right))>\eta$ for every $j\in \{1,2,\cdots ,n\}$.

Definition 3.

The system $(E,h_{1,\infty })$ is said to be an ergodically synchronous sensitive system with the synchronous sensitivity constant η$(\eta >0)$ if for every integer $n>0$, any $a_1,a_2,\cdots ,a_n\in E$ with $a_k\ne a_l$ for any $k,l\in \{1,2,\cdots ,n\}$ with $k\ne l$ and any $\epsilon >0$, there are $b_1,b_2,\cdots ,b_n\in E$ with $b_k\ne b_l$ for any $k,l\in \{1,2,\cdots ,n\}$ with $k\ne l$, satisfying the following:

(1)

$d(a_j,b_j)<\epsilon$ for any $1\le j\le n$;

(2)

One can find $P\subset \{1,2,\cdots \}$, such that $\overline{\mu }\left(P\right)>0$, and that for every $k\in P$, one has that

$$d(h_1^k\left(a_j\right),h_1^k\left(b_j\right))>\eta$$

for every $j\in \{1,2,\cdots ,n\}$.

Definition 4.

The system $(E,h_{1,\infty })$ is said to be ergodically multi-sensitive if there is an $\eta >0$, such that $\overline{\mu }({\displaystyle \bigcap _{j=1}^k}{N}_{h_{1,\infty }}(A_j,\eta ))>0$ for each integer $k\ge 2$ and any open subsets $A_1,A_2,\cdots ,A_k\subset E$ with $A_i\ne Ø$ for each $i\in \{1,2,\cdots ,k\}$.

Definition 5.

The system $(E,h_{1,\infty })$ is said to exhibit upper density one sensitivity if there is an $\eta >0$, such that $\overline{\mu }\left(N_{h_{1,\infty }}(A,\eta )\right)=1$ for any nonempty open subset $A\subset E$.

Definition 6.

The system $(E,h_{1,\infty })$ is said to exhibit density-one sensitive if there exists an $\eta >0$ satisfying that $\mu \left(N_{h_{1,\infty }}(B,\eta )\right)=1$ for every nonempty open subset $B\subset E$.

In particular, if $h_i=h$$(i\in {\mathbb{Z}}^{+})$, one can rewrite the Definitions 1–5 in A.D.D.S. $(E,h)$.

3. Main Results

Let $(E,h_{1,\infty })$ be a N.D.D.S. Assume that $h_j$ is semi-open and satisfies $h_j\circ h=h\circ h_j$ for each $j\in \{1,2,\cdots \}$. Moreover, there is a subset $A\subset \{1,2,\cdots \}$, such that $\mu \left(A\right)=1$ and

$$\underset{j\in A,j\to \infty }{lim}\sum _{i=1}^{j}D(h_i,h)$$

exists. Inspired by [3,11,21], the following establishes the relationship between ergodicity and sensitivity between $(E,h_{1,\infty })$ and $(E,h)$.

First, some results are recalled (Lemmas 1 and 2).

Lemma 1

([20,21,23]). Suppose that $(E,h_{1,\infty })$ is a N.D.D.S., and that h is a continuous map on E. If $h_j\circ h=h\circ h_j$ for each $j\in \{1,2,\cdots \}$, then

(1)

$d(h_1^s\left(y\right),h^s\left(y\right))\le {\displaystyle \sum _{j=1}^s}D(h_j,h)$ for every $y\in E$ and every integer $s\ge 0$;

(2)

$d(h_1^{s+m}\left(y\right),h^m\left(h_1^s\left(y\right)\right))\le {\displaystyle \sum _{j=s+1}^{m+s}}D(h_j,h)$ for every $y\in E$, every integer $m\ge 1$ and every integer $s\ge 1$.

Lemma 2

([24]). Let $P,Q\subset {\mathbb{Z}}^{+}$.

(1)

If $\mu \left(P\right)=1$ and $\overline{\mu }\left(Q\right)=1$, then $\overline{\mu }(P\cap Q)=1$;

(2)

If $\mu \left(P\right)=1$ and $\underline{\mu }\left(Q\right)>0$, then $\overline{\mu }(P\cap Q)>0$.

Theorem 1.

(1)

$(E,h)$ is topologically strong-ergodic if and only if $(E,h_{1,\infty })$;

(2)

$(E,h)$ is upper density-one sensitive if and only if $(E,h_{1,\infty })$;

(3)

$(E,h)$ is density-one sensitive if and only if $(E,h_{1,\infty })$.

Proof.

(1) Assume that $(E,h)$ is topologically strong-ergodic. For any $ϵ>0$, let $U=B(a,ϵ)=\{x\in E:d(a,x)<ϵ\}$ and $V=B(b,ϵ)=\{x\in E:d(b,x)<ϵ\}$. Therefore, both $U\subset E$ and $V\subset E$ are open sets with $U\ne Ø$ and $V\ne Ø$. As there is a subset $A\subset \{1,2,\cdots \}$ such that $\mu \left(A\right)=1$ and

$$\underset{j\in A,j\to \infty }{lim}\sum _{i=1}^{j}D(h_i,h)$$

exists, by Lemma 1(2) and Lemma 2(1), there exists a positive integer $s>0$, which satisfies that for each $ϵ>0$,

$$d(h_1^{s+m}\left(b\right),h^m\left(h_1^s\left(b\right)\right))<\frac{1}{2}ϵ$$

for any integer $m\ge 1$ with $m+s\in A$ and any $b\in Y$. Assume that $U^{\prime }$ denotes the interior of $h_1^s\left(U\right)$, and let $V^{\prime }=B(b,\frac{1}{2}ϵ)$. Then $U^{\prime }\subset E$ and $V^{\prime }\subset E$ are open sets with $U^{\prime }\ne Ø$ and $V^{\prime }\ne Ø$. Since $(E,h)$ is topologically strong-ergodic, then

$$\overline{\mu }\left(N_h(U^{\prime },V^{\prime })\right)=1.$$

For any $m\in N_h(U^{\prime },V^{\prime })$, there exists an $f^{\prime }\in U^{\prime }$ with $h^m\left(f^{\prime }\right)\in V^{\prime }$. Moreover, because $U^{\prime }$ is the interior of $h_1^s\left(U\right)$, there is an $f\in U$ satisfying $f^{\prime }=h_1^s\left(f\right)$. This means that $h^m\left(h_1^s\left(f\right)\right)\in V^{\prime }$. By Lemma 1(2) and Lemma 2(1), one has that

$$d(h_1^{m+s}\left(f\right),h^m\left(h_1^s\left(f\right)\right))\le \sum _{j=s}^{\infty }D(h_j,h)<\frac{1}{2}ϵ$$

for any $m\in N_h(U^{\prime },V^{\prime }):m+s\in A$. So,

$$d(b,h_1^{m+s}\left(u\right))\le d(b,h^m\left(h_1^s\left(u\right)\right))+d(h^m\left(h_1^s\left(u\right)\right),h_1^{m+s}\left(u\right))<ϵ$$

for any $m\in N_h(U^{\prime },V^{\prime }):m+s\in A$. This means that $h_1^{m+s}\left(U\right)\cap V\ne Ø$ for any $m\in N_h(U^{\prime },V^{\prime })$ with $m+s\in A$. That is, $m+s\in N_{h_{1,\infty }}(U,V)$. Hence,

$$[(N_h(U^{\prime },V^{\prime })+s)\cap A]\subset N_{h_{1,\infty }}(U,V).$$

Since $(E,h)$ is topologically strong-ergodic, by the definition, $(E,h_{1,\infty })$ is topologically strong-ergodic.

Now, assume that $(E,h_{1,\infty })$ is topologically strong-ergodic, for any $ϵ>0$, write $V=B(b,\frac{1}{2}ϵ)$, where

$$B(b,\frac{1}{2}ϵ)=\{y\in Y:d(b,y)<\frac{1}{2}ϵ\}.$$

Then $V\subset E$ is open with $V\ne Ø$. Since there is a subset $A\subset \{1,2,\cdots \}$ such that $\mu \left(A\right)=1$ and

$$\underset{j\in A,j\to \infty }{lim}\sum _{i=1}^{j}D(h_i,h)$$

exists, by Lemma 1(2) and Lemma 2(1), there exists some integer $s\ge 1$ with

$$d(h_1^{s+m}\left(b\right),h^m\left(h_1^s\left(b\right)\right))<\frac{1}{2}ϵ$$

for any $ϵ>0$, any integer $m\ge 1:m+s\in A$, and any $b\in E$. Let $U={\left(h_1^s\right)}^{-1}B(a,ϵ)$. Then $U\subset E$ is open with $U\ne Ø$. As $(E,h_{1,\infty })$ is topologically strongly ergodic, then

$$\overline{\mu }\left(N_{h_{1,\infty }}(U,V)\right)=1.$$

For every $m\in N_{h_{1,\infty }}(U,V)$ and any $u\in U$ satisfying $h_1^{m+s}\left(u\right)\in V$, by Lemma 1(2),

$$d(h_1^{m+s}\left(u\right),h^m\left(h_1^s\left(u\right)\right))\le \sum _{j=s}^{\infty }D(h_j,h)<\frac{1}{2}ϵ$$

for any $m\in N_{h_{1,\infty }}(U,V):m+s\in A$ and any $u\in U:h_1^{m+s}\left(u\right)\in V$. Therefore,

$$d(b,h^m\left(h_1^s\left(u\right)\right))\le d(b,h_1^{m+s}\left(u\right)))+d(h^m\left(h_1^s\left(u\right)\right),h_1^{m+s}\left(u\right))<ϵ.$$

This means that $h^m\left(B(a,ϵ)\right)\cap B(b,ϵ)\ne Ø$ for any $m\in N_{h_{1,\infty }}(U,V):m+s\in A$. So, $m+s\in N_{h_{1,\infty }}(U,V)$ implies $m\in N_h(B(a,ϵ),B(b,ϵ)):m+s\in A$. So

$$[(N_{h_{1,\infty }}(U,V)\cap A)-s]\subset N_h(B(a,ϵ),B(b,ϵ)),$$

where $I-s=\{k-s:k\in I,s\le k\}$ for a set $I\subset \{0,1,\cdots \}$.

Thus, $(E,h)$ is topologically strong-ergodic.

(2) Assuming that $(E,h)$ is upper density-one sensitive with a sensitivity constant $\eta >0$, and that $ϵ>0$ is any given. We can write $U=B(a,ϵ)$. Since

$$\sum _{j=1}^{\infty }D(h_j,h)$$

exists, by Lemma 1(2) and Lemma 2(1), there exists some integer $s\ge 1$ satisfying

$$d(h_1^{s+m}\left(b\right),h^m\left(h_1^s\left(b\right)\right))<\frac{1}{2}ϵ$$

for any $ϵ>0$, any integer $m\ge 1:m+s\in A$, and any $b\in E$. Pick $k\in \{1,2,\cdots \}$ with $k>\frac{4}{\eta }$. Since there is a subset $A\subset \{1,2,\cdots \}$ such that $\mu \left(A\right)=1$ and

$$\underset{j\in A,j\to \infty }{lim}\sum _{i=1}^{j}D(h_i,h)$$

exists, there exists some integer $s_0\ge 1$ with

$$d(h_1^{s_0+m}\left(b\right),h^m\left(h_1^{s_0}\left(b\right)\right))<\frac{1}{k}$$

for every integer $m\ge 1:m+s\in A$ and any $b\in E$. Moreover, because $h_j$ is semi-open for every positive integer j, the interior of $h_1^{s_0}\left(U\right)$ is not empty. Assume that $U^{\prime }$ denotes the interior of $h_1^{s_0}\left(U\right)$. Since $(E,h)$ is upper density-one sensitive with $\eta >0$ as a constant of sensitivity, $\overline{\mu }\left(N_h(U^{\prime },\eta )\right)=1$. Let $m\in N_h(U^{\prime },\eta ):m+s_0\in A$. By the definition, there are $v_1,v_2\in h_1^{s_0}\left(U\right)$ satisfying $d(h^m\left(v_1\right),h^m\left(v_2\right))>\eta$. Note that $v_1,v_2\in h_1^{s_0}\left(U\right)$, then there are $v_1^{\prime },v_2^{\prime }\in U$ satisfying $v_1=h_1^{s_0}\left(v_1^{\prime }\right)$, $v_2=h_1^{s_0}\left(v_2^{\prime }\right)$ and $d(h^m\left(h_1^{s_0}\left(v_1^{\prime }\right)\right),h^m\left(h_1^{s_0}\left(v_2^{\prime }\right)\right))>\eta$. Moreover, because

$$d(h_1^{s_0+m}(v_j^{\prime }),h^m(h_1^{s_0}(v_j^{\prime })))<\frac{1}{k}$$

for every $j\in \{1,2\}$, $d(h_1^{s_0+m}\left(v_1^{\prime }\right),h_1^{s_0+m}\left(v_2^{\prime }\right))>\eta -\frac{2}{k}>\frac{1}{2}\eta$. This implies that $s_0+m\in N_{h_{1,\infty }}(U,\frac{1}{2}\eta )$. So,

$$[(N_h(U^{\prime },\eta )+s_0)\cap A]\subset N_{h_{1,\infty }}(U,\frac{1}{2}\eta ).$$

Since $(E,h)$ is upper density-one sensitive, by the definition, $(E,h_{1,\infty })$ is upper density-one sensitive with $\frac{1}{2}\eta$ as a constant of sensitivity.

Now we assume that $(E,h_{1,\infty })$ is upper density-one sensitive with a sensitivity constant $\eta >0$, and that $U\subset E$ is open with $U\ne Ø$. Pick $k\in \{1,2,\cdots \}$ such that $k>\frac{4}{\eta }$. Since there is a subset $A\subset \{1,2,\cdots \}$ such that $\overline{\mu }\left(A\right)=1$ and

$$\underset{j\in A,j\to \infty }{lim}\sum _{i=1}^{j}D(h_i,h)$$

exists, by Lemma 1(2) and Lemma 2(1), there exists some integer $s_0\ge 1$ with

$$d(h_1^{s_0+m}\left(b\right),h^m\left(h_1^{s_0}\left(b\right)\right))<\frac{1}{k}$$

for any integer $m\ge 1$ with $s_0+m\in A$ and any $b\in E$. Since $(E,h_{1,\infty })$ is upper density-one sensitive with a sensitivity constant $\eta >0$, then

$$\overline{D}\left(N_{h_{1,\infty }}(U,\eta )\right)=1$$

and

$$\overline{D}(N_{h_{1,\infty }}({(h_1^{s_0})}^{-1}(U),\eta ))=1.$$

Let m be an integer satisfying $s_0+m\in A$ and $m+s_0\in N_{h_{1,\infty }}({\left(h_1^{s_0}\right)}^{-1}\left(U\right),\eta ).$ Then there are $v_1,v_2\in {\left(h_1^{s_0}\right)}^{-1}\left(U\right)$ with

$$d(h_1^{s_0+m}\left(v_1\right),h_1^{s_0+m}\left(v_2\right))>\eta .$$

Since $v_1,v_2\in {\left(h_1^{s_0}\right)}^{-1}\left(U\right)$, there are $v_1^{\prime },v_2^{\prime }\in U$ satisfying $v_1^{\prime }=h_1^{s_0}\left(v_1\right)$ and $v_2^{\prime }=h_1^{s_0}\left(v_2\right)$. Therefore,

$$d(h_{s_0+m}\circ \cdots \circ h_{s_0+1}\left(v_1^{\prime }\right),h_{s_0+m}\circ \cdots \circ h_{s_0+1}\left(v_2^{\prime }\right))>\eta .$$

Since

$$d(h_1^{s_0+m}\left(v_j\right),h^m\left(h_1^{s_0}\left(v_j\right)\right))<\frac{1}{k}(j\in \{1,2\})\mathrm{and}d(h_1^{s_0+m}(v_j),h^m(v_j^{\prime }))<\frac{1}{k}(j\in \{1,2\}),$$

then

$$d(h^m\left(v_1^{\prime }\right),h^m\left(v_2^{\prime }\right))>\eta -\frac{2}{k}>\frac{1}{2}\eta .$$

So $m\in N_h(U,\frac{1}{2}\eta )$. Consequently, one has

$$[N_{h_{1,\infty }}({\left(h_1^{s_0}\right)}^{-1}\left(U\right),\eta )\cap A-s_0]\subset N_h(U,\frac{1}{2}\eta ).$$

Thus, $(E,h)$ is upper density-one sensitive.

(3) The proof is similar to that of (2). □

Theorem 2.

(1) If $(E,h)$ satisfies that $\underline{\mu }\left(N_h(A,B)\right)>0$ for every pair of open subsets $A,B\subset E$ with $A,B\ne Ø$, then $(E,h_{1,\infty })$ is topologically ergodic.

(2) If $(E,h_{1,\infty })$ satisfies that $\underline{\mu }\left(N_{h_{1,\infty }}(A,B)\right)>0$ for every pair of open subsets $A,B\subset E$ with $A,B\ne Ø$, then $(E,h)$ is topologically ergodic.

Proof.

(1) Assume that $(E,h)$ satisfies that $\underline{\mu }\left(N_h(A,B)\right)>0$ for every pair of open subsets $A,B\subset E$ with $A,B\ne Ø$. For every $ϵ>0$, let

$$U=B(a,ϵ)=\{y\in E:d(a,y)<ϵ\}$$

and

$$V=B(b,ϵ)=\{y\in E:d(b,y)<ϵ\}.$$

Therefore, both $U\subset E$ and $V\subset E$ are open with $U,V\ne Ø$. As there is a subset $A\subset \{1,2,\cdots \}$ such that $\mu \left(A\right)=1$ and

$$\underset{j\in A,j\to \infty }{lim}\sum _{i=1}^{j}D(h_i,h)$$

exists, by Lemma 1(2) and Lemma 2(1), there exists some positive integer $s>0$ satisfying that for every $ϵ>0$,

$$d(h_1^{s+m}\left(b\right),h^m\left(g_1^s\left(b\right)\right))<\frac{1}{2}ϵ$$

for every integer $m\ge 1:m+s\in A$ and every $b\in E$. Assume that $U^{\prime }$ is the interior of $h_1^s\left(U\right)$, and that $V^{\prime }=B(b,\frac{1}{2}ϵ)$. Since $h_j$ is semi-open for every positive integer j, $U^{\prime }\subset E$ and $V^{\prime }\subset E$ are open with $U^{\prime },V^{\prime }\ne Ø$,

$$\underline{\mu }\left(N_h(U^{\prime },V^{\prime })\right)>0.$$

For any $m\in N_h(U^{\prime },V^{\prime })$, there exists an $f^{\prime }\in U^{\prime }$ with $h^m\left(f^{\prime }\right)\in V^{\prime }$. Moreover, because $U^{\prime }$ is the interior of $h_1^s\left(U\right)$, then there is an $f\in U$ with $f^{\prime }=h_1^s\left(f\right)$. This means that $h^m\left(h_1^s\left(f\right)\right)\in V^{\prime }$. By Lemma 1(2) and Lemma 2(1), one has that

$$d(h_1^{m+s}\left(f\right),h^m\left(h_1^s\left(f\right)\right))\le \sum _{j=s}^{\infty }D(h_j,h)<\frac{1}{2}ϵ$$

for any $m\in N_h(U^{\prime },V^{\prime }):m+s\in A$. Therefore,

$$d(b,h_1^{m+s}\left(u\right))\le d(b,h^m\left(h_1^s\left(u\right)\right))+d(h^m\left(h_1^s\left(u\right)\right),h_1^{m+s}\left(u\right))<ϵ$$

for any $m\in N_h(U^{\prime },V^{\prime }):m+s\in A$. This means that $h_1^{m+s}\left(U\right)\cap V\ne Ø$ for any $m\in N_h(U^{\prime },V^{\prime }):m+s\in A$. So, $m+s\in N_{h_{1,\infty }}(U,V)$ for any $m\in N_h(U^{\prime },V^{\prime }):m+s\in A$. Then

$$(N_h(U^{\prime },V^{\prime })+s)\cap A\subset N_{h_{1,\infty }}(U,V).$$

Moreover, since $\underline{\mu }\left(N_h(U^{\prime },V^{\prime })\right)>0$, by Lemma 2(2), $\overline{\mu }\left(N_{h_{1,\infty }}(U,V)\right)>0.$

Thus, $(E,h_{1,\infty })$ is topologically ergodic.

(2) Assume that $(E,h_{1,\infty })$ satisfies that $\underline{\mu }\left(N_{h_{1,\infty }}(A,B)\right)>0$ for every pair of open subsets $A,B\subset E$ with $A,B\ne Ø$. For any given $ϵ>0$, write $V=B(b,\frac{1}{2}ϵ)$, where

$$B(b,\frac{1}{2}ϵ)=\{y\in E:d(b,y)<\frac{1}{2}ϵ\}.$$

Then $V\subset E$ is nonempty and open. Since there is a subset $A\subset \{1,2,\cdots \}$ such that $\mu \left(A\right)=1$ and

$$\underset{j\in A,j\to \infty }{lim}\sum _{i=1}^{j}D(h_i,h)$$

exists, by Lemma 1(2) and Lemma 2(1), there exists some positive integer $s>0$, which satisfies that for every $ϵ>0$,

$$d(g_1^{s+m}\left(b\right),g^m\left(g_1^s\left(b\right)\right))<\frac{1}{2}ϵ$$

for every positive integer m with $m+s\in A$ and every $b\in E$. Let $U={\left(h_1^s\right)}^{-1}\left(B(a,ϵ)\right)$. Then $U\subset E$ is not empty and open. By hypothesis,

$$\underline{\mu }\left(N_{h_{1,\infty }}(U,V)\right)>0,\overline{\mu }(N_{h_{1,\infty }}(U,V)=1.$$

For every $m\in N_{h_{1,\infty }}(U,V)$ and any $u\in U:h_1^{m+s}\left(u\right)\in V$. By Lemma 1(2), one has that

$$d(h_1^{m+s}\left(u\right),h^m\left(h_1^s\left(u\right)\right))\le \sum _{j=s}^{\infty }D(h_j,h)<\frac{1}{2}ϵ$$

for any $m\in N_{h_{1,\infty }}(U,V):m+s\in A$ and any $u\in U:h_1^{m+s}\left(u\right)\in V$. Then

$$d(b,h^m\left(h_1^s\left(u\right)\right))\le d(b,h_1^{m+s}\left(u\right)))+d(h^m\left(h_1^s\left(u\right)\right),h_1^{m+s}\left(u\right))<ϵ$$

for any $m\in N_{h_{1,\infty }}(U,V):m+s\in A$ and any $u\in U:h_1^{m+s}\left(u\right)\in V$. This means that $h^m\left(B(a,ϵ)\right)\cap B(b,ϵ)\ne Ø$ for any $m\in N_{h_{1,\infty }}(U,V):m+s\in A$. So, $m+s\in N_{h_{1,\infty }}(U,V)$ for any $m\in N_g(B(a,ϵ),B(b,ϵ)):m+s\in A$. Thus

$$[(N_{h_{1,\infty }}(U,V)\cap A)-s]\subset N_h(B(a,ϵ),B(b,ϵ)).$$

Moreover, since

$$\underline{\mu }\left(N_{h_{1,\infty }}(U,V)\right)>0,$$

then, by Lemma 2(2),

$$\overline{\mu }\left(N_h(B(a,\alpha ),B(b,\alpha ))\right)>0.$$

Thus, $(E,h)$ is topologically ergodic. □

The following will discuss some stronger forms of ergodic sensitivity. For the notion of ergodically sensitive, we refer the reader to [23]. The definitions of ergodically multi-sensitive, ergodically synchronously sensitive, and ergodically collectively sensitive see Definitions 1–3.

Theorem 3.

(1) Assume that $(E,h)$ satisfies that $\underline{\mu }({\displaystyle \bigcap _{j=1}^k}{N}_h(A_j,\eta ))>0$ for some $\eta >0$, every positive integer $k>1$ and any open subsets $A_1,A_2,\cdots ,A_k\subset E$ with $A_1,A_2,\cdots ,A_k\ne Ø$. Then $(E,h_{1,\infty })$ is ergodically multi-sensitive.

(2) Assume that $(E,h_{1,\infty })$ satisfies that $\underline{\mu }({\displaystyle \bigcap _{j=1}^k}{N}_{h_{1,\infty }}(A_j,\eta ))>0$ for $\eta >0$, every positive integer $k>1$ and any open subsets $A_1,A_2,\cdots ,A_k\subset E$ with $A_1,A_2,\cdots ,A_k\ne Ø$. Then $(E,h)$ is ergodically multi-sensitive.

Proof.

(1) Suppose that $\underline{\mu }({\displaystyle \bigcap _{j=1}^k}{N}_h(A_j,\eta ))>0$ for some $\eta >0$, every positive integer $k>1$ and any open subsets $A_1,A_2,\cdots ,A_k\subset E$ with $A_1,A_2,\cdots ,A_k\ne Ø$.

Since there is a subset $A\subset \{1,2,\cdots \}$ such that $\mu \left(A\right)=1$ and

$$\underset{j\in A,j\to \infty }{lim}\sum _{i=1}^{j}D(h_i,h)$$

exists, by Lemma 1(2) and Lemma 2(2), there exists some positive integer $s>0$ which satisfies that for every $ϵ>0$,

$$d(h_1^{s+m}\left(b\right),h^m\left(g_1^s\left(b\right)\right))<\frac{1}{2}ϵ$$

for every positive integer m with $m+s\in A$ and every $b\in E$. Pick $t\in \{1,2,\cdots \}$ satisfying $t>\frac{4}{\eta }$. Then there exists some positive integer $s_0$ with

$$d(h_1^{s_0+m}\left(b\right),h^m\left(g_1^{s_0}\left(b\right)\right))<\frac{1}{t}$$

for every positive integer m with $m+s_0\in A$ and every $b\in E$.

Let $U_j\in E(j\in \{1,2,\cdots ,k\})$ be nonempty open sets. Since $h_j$ is semi-open for every positive integer j, the interior of $h_1^{s_0}\left(U_j\right)$ is nonempty for every $j\in \{1,2,\cdots ,k\}$. Assume that $U_j^{\prime }$ denotes the interior of $h_1^{s_0}\left(U_j\right)$ for every $j\in \{1,2,\cdots ,k\}$. By hypothesis,

$$\underline{\mu }(\bigcap _{j=1}^k{N}_h(U_j^{\prime },\lambda ))>0.$$

Let $m\in {\displaystyle \bigcap _{j=1}^k}{N}_h(U_j^{\prime },\eta )$ and $m+s_0\in A$. By the definition, there are $u_j,v_j\in h_1^{s_0}\left(U_j\right)$, which satisfy that $d(h^m\left(u_j\right),h^m\left(v_j\right))>\eta$ for every $j\in \{1,2,\cdots ,k\}$. Since $u_j,v_j\in h_1^{s_0}\left(U_j\right)$ for every $j\in \{1,2,\cdots ,k\}$, there exist $u_j^{\prime },v_j^{\prime }\in U$ satisfying $u_j=h_1^{s_0}(u_j^{\prime })$ for every $j\in \{1,2,\cdots ,k\}$ and $d(h^m(h_1^{s_0}(u_j^{\prime })),h^m(h_1^{s_0}(v_j^{\prime })))>\eta$ for every $j\in \{1,2,\cdots ,k\}$. It is clear that

$$d(h_1^{s_0+m}(u_j^{\prime }),h^m(h_1^{s_0}(u_j^{\prime })))<\frac{1}{t}$$

and

$$d(h_1^{s_0+m}(v_j^{\prime }),h^m(h_1^{s_0}(v_j^{\prime })))<\frac{1}{t}$$

for every $j\in \{1,2,\cdots ,k\}$. Thus, $d(h_1^{s_0+m}(u_j^{\prime }),h_1^{s_0+m}(v_j^{\prime }))>\eta -\frac{2}{t}>\frac{1}{2}\eta$, which means that $s_0+m\in \bigcap _{j=1}^k{N}_{h_{1,\infty }}(U_j,\frac{1}{2}\eta )$. So,

$$[(\bigcap _{j=1}^m{N}_h(U_j^{\prime },\eta )+s_0)\cap A]\subset \bigcap _{j=1}^k{N}_{h_{1,\infty }}(U_j,\frac{1}{2}\eta ).$$

Thus, $(E,h_{1,\infty })$ is ergodically multi-sensitive, where $\frac{1}{2}\eta$ is a constant of sensitivity.

(2) Assume that $\underline{\mu }({\displaystyle \bigcap _{j=1}^n}{N}_{h_{1,\infty }}(A_j,\eta ))>0$ for $\eta >0$, every positive integer $k>1$ and any open subsets $A_1,A_2,\cdots ,A_n\subset E$ with $A_1,A_2,\cdots ,A_k\ne Ø$. Pick $k\in \{1,2,\cdots \}$ satisfying $k>\frac{4}{\eta }$. Since there is a subset $A\subset \{1,2,\cdots \}$ such that $\mu \left(A\right)=1$ and

$$\underset{j\in A,j\to \infty }{lim}\sum _{i=1}^{j}D(h_i,h)$$

exists, by Lemma 1(2) and Lemma 2(2), there exists a positive integer s with

$$d(h_1^{s+m}\left(b\right),h^m\left(h_1^s\left(b\right)\right))<\frac{1}{k}$$

for every positive integer m with $m+s\in A$ and every $b\in E$. Let $U_j\subset E(j=1,2,\cdots ,n;n\ge 2)$ be nonempty open sets and $u_j\in U_j$ for every $j\in \{1,2,\cdots ,n\}$. For any $ϵ>0$, one can pick an integer $r>0$ with $\frac{1}{r}<ϵ$. This means that $\frac{1}{m}<ϵ$ for every positive integer $m\ge r$. By hypothesis,

$$\underline{\mu }(\bigcap _{j=1}^n{N}_{h_{1,\infty }}(U_j,\eta ))>0.$$

Since $h_j$ is continuous for each $j\in \{1,2,\cdots \}$, then ${\left(h_1^s\right)}^{-1}\left(U_j\right)$ is open with ${\left(h_1^s\right)}^{-1}\left(U_j\right)\ne Ø$ for every $j\in \{1,2,\cdots \}$. By hypothesis,

$$\underline{\mu }(\bigcap _{j=1}^n{N}_{h_{1,\infty }}({\left(h_1^s\right)}^{-1}\left(U_j\right),\eta ))>0.$$

For any

$$m+s\in \bigcap _{j=1}^n{N}_{h_{1,\infty }}({\left(h_1^s\right)}^{-1}\left(U_j\right),\eta )$$

with $m+s\in A$, there are

$$y_j,z_j\in N_{h_{1,\infty }}({\left(h_1^s\right)}^{-1}\left(U_j\right),\eta )$$

satisfying

$$d(h_1^{s+m}\left(y_j\right),h_1^{s+m}\left(z_j\right))>\eta (j\in \{1,2,\cdots ,n\}).$$

Since

$$y_j,z_j\in N_{h_{1,\infty }}{\left(h_1^s\right)}^{-1}\left(U_j\right),\eta )(j\in \{1,2,\cdots ,n\}),$$

then there exists $y_j^{\prime },z_j^{\prime }\in U_j:y_j^{\prime }=h_1^s\left(y_j\right)$, $z_j^{\prime }=h_1^s\left(z_j\right)$ such that

$$d(h_{s+m}\circ h_{s+1}(y_j^{\prime }),h_{s+m}\circ h_{s+1}(z_j^{\prime }))>\eta (j\in \{1,2,\cdots ,n\}).$$

It is clear that

$$d(h_1^{s+m}\left(y_j\right),h^m\left(h_1^s\left(y_j\right)\right))<\frac{1}{k},i.e.,d(h_1^{s+m}\left(y_j\right),h^m(y_j^{\prime }))<\frac{1}{k}$$

for every positive integer m with $m+s\in A$ and every $j\in \{1,2,\cdots ,n\}$. By the above argument and triangle inequality,

$$d(h^m\left(y_j^{\prime }\right),h^m\left(z_j^{\prime }\right))>\eta -\frac{2}{k}>\frac{1}{2}\eta$$

for every $j\in \{1,2,\cdots ,n\}$. So $d(h^m\left(y_j^{\prime }\right),h^m\left(z_j^{\prime }\right))>\frac{1}{2}\eta$ for any $y_j^{\prime },z_j^{\prime }\in U_j(j=1,2,\cdots ,n)$. This means that

$$m\in \bigcap _{j=1}^n{N}_h(U_j,\frac{1}{2}\eta ).$$

Consequently,

$$[(\bigcap _{j=1}^n{N}_{h_{1,\infty }}({\left(h_1^{s_0}\right)}^{-1}\left(U_j\right),\eta )\cap A)-s]\subset \bigcap _{j=1}^n{N}_h(U_j,\frac{1}{2}\eta ).$$

Thus, $(E,h)$ is ergodically multi-sensitive. □

Theorem 4.

(1) If $(E,h)$ satisfies that $\underline{\mu }(N_h(A,\eta ))>0$ for some $\eta >0$ and every open subset $A\subset E$ with $A\ne Ø$, then $(E,h_{1,\infty })$ is ergodically sensitive.

(2) If $(E,h_{1,\infty })$ satisfies that $\underline{\mu }(N_{h_{1,\infty }}(A,\eta ))>0$ for some $\eta >0$ and every open subset $A\subset E$ with $A\ne Ø$, then $(E,h)$ is ergodically sensitive.

Proof. 

Let $k=1$ in Theorem 3, the conclusions are obtained. □

Theorem 5.

(1) For some $\eta >0$, any finitely many distinct points $a_1,a_2,\cdots ,a_n$ of E and every $ϵ>0$, if there are $b_1,b_2,\cdots ,b_n$ of E with $b_k\ne b_l$ for any $k,l\in \{1,2,\cdots ,n\}$ with $k\ne l$, which satisfy the following conditions.

(i) $d(a_j,b_j)<ϵ$ for every $j\in \{1,2,\cdots ,n\}$;

(ii) There exists some $B\subset \{1,2,\cdots \}$ with $\underline{\mu }(B)>0$, such that $d(h^k(a_j),h^k(b_j))\ge ϵ$ for every $j\in \{1,2,\cdots ,n\}$ and every $k\in B$,

then $(E,h_{1,\infty })$ is ergodically synchronous sensitive.

(2) For some $\eta >0$, any finitely many distinct points $a_1,a_2,\cdots ,a_n$ of E and any $ϵ>0$, there are $b_1,b_2,\cdots ,b_n$ of E with $b_k\ne b_l$ for any $k,l\in \{1,2,\cdots ,n\}:k\ne l$, which satisfy the following conditions.

(i) $d(a_j,b_j)<ϵ$ for every $j\in \{1,2,\cdots ,n\}$;

(ii) There exists some $B\subset \{1,2,\cdots \}$ with $\underline{\mu }(B)>0$, such that $d(h_1^k(a_j),h_1^k(b_j))\ge \eta$ for every $j\in \{1,2,\cdots ,n\}$ and every $k\in B$,

then $(E,h)$ is ergodically synchronous sensitive.

Proof.

(1) Let $a_1,a_2,\cdots ,a_n\in E:a_i\ne a_j(i\ne j)$ for any $i,j\in \{1,2,\cdots ,n\}$ and $ϵ>0$ be given, and let $V_j=B(a_j,ϵ)$ for every $j\in \{1,2,\cdots ,n\}$. Since

$$\underset{j\in A,j\to \infty }{lim}\sum _{i=1}^{j}D(h_i,h)$$

exists, by Lemma 1(2) and Lemma 2(2), there exists a positive integer s satisfying that for every $ϵ>0$,

$$d(h_1^{s+m}(b),h^m(h_1^s(b)))<ϵ$$

for every positive integer m with $m+s\in A$ and every $b\in E$. Pick $k\in \{1,2,\cdots \}$ satisfying $k>\frac{4}{\eta }$. Then there exists a positive integer $s_0$ satisfying

$$d(h_1^{s_0+m}(b),h^m(h_1^{s_0}(b)))<\frac{1}{k}$$

for every positive integer m with $m+s_0\in A$ and every $b\in E$.

Let $v_j=h_1^{s_0}(a_j)\in h_1^{s_0}(V_j)$ for every $j\in \{1,2,\cdots ,n\}$. Then, there exists $u_1,u_2,\cdots ,u_n$ of E with $u_k\ne u_l$ for any $k,l\in \{1,2,\cdots ,n\}:k\ne l$, which satisfy the following conditions.

(i) $d(v_j,u_j)<ϵ$ for every $j\in \{1,2,\cdots ,n\}$;

(ii) There exists some $B\subset \{1,2,\cdots \}$ with $\underline{\mu }\left(B\right)>0$, such that $d(h^t\left(v_j\right),h^t\left(u_j\right))\ge \eta$ for every $j\in \{1,2,\cdots ,n\}$ and every $t\in B$.

As $h_j$ is semi-open for every $j\in \{1,2,\cdots \}$, the interior of $h_1^{s_0}\left(V_j\right)$ is nonempty for every $j\in \{1,2,\cdots ,n\}$. So,

$$u_j\in B(v_j,ϵ)=B(h_1^{s_0}\left(a_j\right),ϵ)\subset h_1^{s_0}\left(V_j\right)$$

for any $j\in \{1,2,\cdots ,n\}$. This means that there are $b_j\in V_j$ with $u_j=h_1^{s_0}\left(b_j\right)$ for every $j\in \{1,2,\cdots ,n\}$. Since $V_j=B(a_j,ϵ)$ for every $j\in \{1,2,\cdots ,n\}$, by Lemma 1(2), Lemma 2(2), and the triangle inequality, we have $d(h_1^{s_0+m}\left(a_j\right),h_1^{s_0+m}\left(b_j\right))>\eta -\frac{2}{k}>\frac{1}{2}\eta$ for every positive integer m with $m+s_0\in A$ and any $j\in \{1,2,\cdots ,n\}$; thus, $(E,h_{1,\infty })$ is ergodically synchronously sensitive with $\frac{1}{2}\eta$ as a constant of sensitivity.

(2) Let $a_1,a_2,\cdots ,a_n\in E:a_k\ne a_l(k\ne l)$ for any $k,l\in \{1,2,\cdots ,n\}$ and $ϵ>0$ be given, and let $V_i=B(a_i,ϵ)$ for each $i\in \{1,2,\cdots ,n\}$. We pick $k\in \{1,2,\cdots \}$ satisfying $k>\frac{4}{\eta }$. There exists a positive integer $s_0=$ with

$$d(h_1^{s_0+m}\left(b\right),h^m\left(h_1^{s_0}\left(b\right)\right))<\frac{1}{k}$$

for every positive integer m with $m+s_0\in A$ and every $b\in E$. Let

$$v_i\in {\left(h_1^i\right)}^{-1}\left(a_i\right)\subset {\left(h_1^i\right)}^{-1}\left(V_i\right)$$

for any $i\in \{1,2,\cdots ,n\}$. Then, there exists $u_j\in B(v_j,ϵ)$ satisfying $u_j\in {\left(h_1^j\right)}^{-1}\left(V_j\right)$ and

$$d(h_1^{s_0+m}\left(v_j\right),h_1^{s_0+m}\left(u_j\right))\ge \eta$$

for each integer $1\le j\le n$ and every $m\in B$. This means that there exists $b_j\in V_j$ satisfying $b_j=h_1^{s_0}\left(u_j\right)$ for any $j\in \{1,2,\cdots ,n\}$. So,

$$d(h_{s_0+m}\circ \cdots \circ h_{s_0+1}\left(a_j\right),h_{s_0+m}\circ \cdots \circ h_{s_0+1}\left(b_j\right))\ge \eta$$

for every integer $1\le j\le n$ and every $m\in B$. Thus, it can be seen that

$$d(h_1^{s_0+m}\left(u_j\right),h^m\left(b_j\right))<\frac{1}{k}$$

for any integer $j\in \{1,2,\cdots ,n\}$ and every $m\in B$, one can obtain

$$d(h_1^{s_0+m}\left(v_j\right),h^m\left(a_j\right))<\frac{1}{k}$$

for each $j\in \{1,2,\cdots ,n\}$ and every $m\in B$. Since $V_j=B(a_j,ϵ)$ for any $j\in \{1,2,\cdots ,n\}$, by the hypothesis, Lemma 1(2), Lemma 2(2), and triangle inequality $d(h^m\left(a_j\right),h^m\left(b_j\right))>\eta -\frac{2}{k}>\frac{1}{2}\eta$ for any positive integer $m>0:m+s_0\in A$, every $j\in \{1,2,\cdots ,n\}$, $(E,h)$ is ergodically synchronous sensitive, where $\frac{1}{2}\eta$ is a constant of sensitivity. □

Theorem 6.

(1) For some $\eta >0$, any finitely many distinct points $a_1,a_2,\cdots ,a_n$ of E and every $ϵ>0$, there exist $b_1,b_2,\cdots ,b_n$ of E, such that $b_k\ne b_l$ for any $k,l\in \{1,2,\cdots ,n\}:k\ne l$, which satisfy the following conditions.

(i) $d(a_j,b_j)<ϵ$ for any $j\in \{1,2,\cdots ,n\}$;

(ii) There are $j_0\in \{1,2,\cdots ,n\}$ and some $B\subset \{1,2,\cdots \}$ with $\underline{\mu }\left(B\right)>0$ with $d(h^k\left(a_{j_0}\right),h^k\left(b_j\right))\ge \eta$ or $d(h^k\left(a_j\right),h^k\left(b_{j_0}\right))\ge \eta$ for every $j\in \{1,2,\cdots ,n\}$ and every $k\in B$, then $(E,h_{1,\infty })$ is ergodically collectively sensitive.

(2) For some $\eta >0$ and any finitely many distinct points $a_1,a_2,\cdots ,a_n$ of E and any $ϵ>0$, there exist $b_1,b_2,\cdots ,b_n$ of E with $b_k\ne b_l$ for any $k,l\in \{1,2,\cdots ,n\}:k\ne l$, which satisfy the following:

(i) $d(a_j,b_j)<ϵ$ for any $j\in \{1,2,\cdots ,n\}$;

(ii) There are $j_0\in \{1,2,\cdots ,n\}$ and some $B\subset \{1,2,\cdots \}$ with $\underline{\mu }\left(B\right)>0$ with $d(h_1^k\left(a_{j_0}\right),h_1^k\left(b_j\right))\ge \eta$ or $d(h_1^k\left(a_j\right),h_1^k\left(b_{j_0}\right))\ge \eta$ for every $j\in \{1,2,\cdots ,n\}$ and every $k\in B$,

then $(E,h)$ is ergodically collectively sensitive.

Proof. 

The proof is similar to that of Theorem 5. □

Example 1.

Let $h_1\left(a\right)=a$ for any $a\in [0,1]$ and $h_j=1-|1-2a|$ for any $a\in [0,1]$, any $j\in \{2,3,\cdots \}$, and let $h=h_2$. Since $h_j$ is the tent map for any $j\in \{2,3,\cdots \}$, by Corollary 4.3 in [25] and the methods of Example 3.2 and Example 4.2 in [20], the sequence ${\left(h_j\right)}_{j=1}^{\infty }$ satisfies the conditions and the conclusions of all theorems in this paper.

Example 2

([26]). Let $g_{2j-1}$ be the logistic map $f\left(a\right)=4a(1-a)$ for any $a\in [0,1]$ and any $j\in \{1,2,\cdots \}$, and for any $j\in \{1,2,\cdots \}$, $h_{2j}$ define $g_{2j}\left(a\right)=3a$ for $a\in \left[0,\frac{1}{3}\right]$, $g_{2j}\left(a\right)=-3a+2$ for $a\in \left[\frac{1}{3},\frac{2}{3}\right]$, and $g_{2j}\left(a\right)=3a-2$ for $a\in \left[\frac{2}{3},1\right]$.

The function images of $g_1^2$, $g_1^3$, and $g_1^4$ are given in Figure 1, and the image of $g_1^n(n>4)$ can be inferred.

Figure 1. The function images of $g_1^2$, $g_1^3$, and $g_1^4$.

From Figure 1, it can be seen that for any $a\in H$ and $\epsilon >0$, there is an $n\in \mathbb{N}$, such that $[0,1]$ is covered by $g_1^n\left(B(a,\epsilon )\right)$. This implies that there exists an $\eta >0$ and a $b\in B(a,\epsilon )$, such that $\rho (g_1^n\left(a\right),g_1^n\left(b\right))>\eta$. Thus, $g_{1,\infty }$ satisfies the sensitivity in this paper.

4. Conclusions

By upper density, lower density, density, a sequence of positive integers, and triangle inequality, some new forms of ergodicity and sensitivity for the system $(E,h_{1,\infty })$ are defined. Given the meanings of ergodicity and sensitivity, this study presents the equivalent system of a nonautonomous discrete dynamical system under some conditions. It can simplify the problem of mapping sequences to the case of single mapping. In fact, sensitivity and ergodicity are the essential features of chaos, which has a wide range of applications in encryption, control, complex networks, etc. The conditions of our results relax and extend the conditions of some responding results in [20], and conclusions of all theorems in this paper enhance those in responding theorems in [20]. So, the results extend and improve the responding ones in [20]. In the future, by using a nonautonomous discrete dynamical system, one can continue to explore new results on T-S fuzzy sampled-data stabilization for switched chaotic systems with its applications.