Some Iterative Properties of $({\mathcal{F}}_1,{\mathcal{F}}_2)$-Chaos in Non-Autonomous Discrete Systems

Abstract

This paper is concerned with invariance $({\mathcal{F}}_1,{\mathcal{F}}_2)$-scrambled sets under iterations. The main results are an extension of the compound invariance of Li–Yorke chaos and distributional chaos. New definitions of $({\mathcal{F}}_1,{\mathcal{F}}_2)$-scrambled sets in non-autonomous discrete systems are given. For a positive integer k, the properties $P(k)$ and $Q(k)$ of Furstenberg families are introduced. It is shown that, for any positive integer k, for any $s\in [0,1]$, Furstenberg family $\overline{M}(s)$ has properties $P(k)$ and $Q(k)$, where $\overline{M}(s)$ denotes the family of all infinite subsets of ${\mathbb{Z}}^{+}$ whose upper density is not less than s. Then, the following conclusion is obtained. D is an $(\overline{M}(s),\overline{M}(t))$-scrambled set of $(X,f_{1,\infty })$ if and only if D is an $(\overline{M}(s),\overline{M}(t))$-scrambled set of $(X,f_{1,\infty }^{\left[m\right]})$.

1. Introduction

Chaotic properties of a dynamical system have been extensively discussed since the introduction of the term chaos by Li and Yorke in 1975 [1] and Devaney in 1989 [2]. To describe some kind of unpredictability in the evolution of a dynamical system, other definitions of chaos have also been proposed, such as generic chaos [3], dense chaos [4], Li–Yorke sensitivity [5], and so on. An important generalization of Li–Yorke chaos is distributional chaos, which is given in 1994 by B. Schweizer and J. Smítal [6]. Then, theories related to scrambled sets are discussed extensively (see [7,8,9,10,11,12] and others). In 1997, the Furstenberg family was introduced by E. Akin [13]. J. Xiong, F. Tan described chaos with a couple of Furstenberg Families. $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos has also been defined [14]. Moreover, $\mathcal{F}$-sensitivity was given in [15] and shadowing properties were discussed in [16]. Most existing papers studied the chaoticity in autonomous discrete systems $(X,f)$. However, if a sequence of perturbations to a system are described by different functions, then there are a sequence of maps to describe them, giving rise to non-autonomous systems. Non-autonomous discrete systems were precisely introduced in [17], in connection with non-autonomous difference equations (see [18,19] and some references therein).

Let $(X,\rho )$ (briefly, X) be a compact metric space and consider a sequence of continuous maps $f_n:X\to X,n\in \mathbb{N}$, denoted by $f_{1,\infty }=(f_1,f_2,\cdots )$. This sequence defines a non-autonomous discrete system $(X,f_{1,\infty })$. The orbit of any point $x\in X$ is given by the sequence $(f_1^n(x))=Orb(x,f_{1,\infty })$, where $f_1^n=f_n\circ \cdots \circ f_1$ for $n\ge 1$, and $f_1^0$ is the identity map.

For $m\in \mathbb{N}$, define

$$g_1=f_m\circ \cdots \circ f_1,g_2=f_{2m}\circ \cdots \circ f_{m+1},\cdots ,g_p=f_{pm}\circ \cdots \circ f_{(p-1)m+1},\cdots .$$

Call $(X,g_{1,\infty })$ a compound system of $(X,f_{1,\infty })$.

Also, denote $g_{1,\infty }$ by $f_{1,\infty }^{\left[m\right]}$ and denote $f_n^k=f_{n+k-1}\circ \cdots \circ f_n$ for $n\ge 1$. By [5], if ${(f_n)}_{n=1}^{\infty }$ converges uniformly to a map f. Then, for any $m\ge 2(m\in \mathbb{N})$, the sequence ${(f_n^{n+m-1})}_{n=1}^{\infty }$ converges uniformly to $f^m$.

In the present work, some notions relating to Furstenberg families and properties $P(k)$, $Q(k)$ are recalled in Section 2 and Section 3. Section 4 states some definitions about $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos. In Section 5, it is proved that, under the conditions of property $P(k)$ and positive shift-invariant, $f_{1,\infty }$ is $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos (strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos, strong $\mathcal{F}$-chaos) implies $f_{1,\infty }^{\left[k\right]}$($k\in {\mathbb{Z}}^{+}$) is $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos (strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos, strong $\mathcal{F}$-chaos). If the conditions property $Q(k)$ and negative shift-invariant both hold, the above conclusion can be inversed. As a conclusion, for arbitrary s and t in $[0,1]$, for every $k\in {\mathbb{Z}}^{+}$, $f_{1,\infty }$ and $f_{1,\infty }^{\left[k\right]}$ can share the same $(\overline{M}(s),\overline{M}(t))$-scrambled set (Theorem 3).

In this paper, it is always assumed that all the maps $f_n$, $n\in \mathbb{N}$, are surjective. It should be noted that this condition is needed by most papers dealing with this kind of system (for example, [20,21,22,23]). It is assumed that sequence ${(f_n)}_{n=1}^{\infty }$ converges uniformly. The aim of this paper is to investigate the $({\mathcal{F}}_1,{\mathcal{F}}_2)$-scrambled sets of $f_{1,\infty }$.

2. Furstenberg Families

Let $\mathcal{P}$ be the collection of all subsets of the positive integers set ${\mathbb{Z}}^{+}=\{0,1,2,\dots \}$. A collection $\mathcal{F}\subset \mathcal{P}$ is called a Furstenberg family if it is hereditary upwards, i.e., $F_1\subset F_2$ and $F_1\in \mathcal{F}$ imply $F_2\in \mathcal{F}$. Obviously, the collection of all infinite subsets of ${\mathbb{Z}}^{+}$ is a Furstenberg family, denoted by $\mathcal{B}$.

Define the dual family $k\mathcal{F}$ of a Furstenberg family $\mathcal{F}$ by

$$k\mathcal{F}=\{F\in \mathcal{P}:{\mathbb{Z}}^{+}-F\notin \mathcal{F}\}=\{F\in \mathcal{P}:F\cap F^{\prime }\ne \varphi \mathrm{for}\mathrm{any}{F}^{\prime }\in \mathcal{F}\}.$$

It is clear that $k\mathcal{F}$ is a Furstenberg family and $k(k\mathcal{F})=\mathcal{F}$ (see [13]).

For $F\in \mathcal{P}$, $i\in {\mathbb{Z}}^{+}$, let $F-i=\{j-i\ge 0:j\in F\}$ and $F+i=\{j+i\ge 0:j\in F\}$. Furstenberg family $\mathcal{F}$ is positive shift-invariant if $F+i\in \mathcal{F}$ for every $F\in \mathcal{F}$ and any $i\in {\mathbb{Z}}^{+}$. Furstenberg family $\mathcal{F}$ is negative shift-invariant if $F-i\in \mathcal{F}$ for every $F\in \mathcal{F}$ and any $i\in {\mathbb{Z}}^{+}$. Furstenberg family $\mathcal{F}$ is shift-invariant if it is positive shift-invariant and negative shift-invariant.

The following shows a class of Furstenberg families which is related to upper density.

Let $F\subset \mathcal{P}$. The upper density and the lower density of F are defined as follows:

$$\overline{\mu }(F)=\underset{n\to \infty }{lim\; sup}\frac{\#(F\cap \{0,1,\dots ,n-1\})}{n},\underset{\_}{\mu }(F)=\underset{n\to \infty }{lim\; inf}\frac{\#(F\cap \{0,1,\dots ,n-1\})}{n},$$

where $\#(A)$ denotes the cardinality of the set A.

For any s in $[0,1]$, set $\overline{M}(s)=\{F\in \mathcal{B}:\overline{\mu }(F)\ge s\}$.

Proposition 1.

For any s in $[0,1]$, $\overline{M}(s)$ is shift-invariant Furstenberg family. And $\overline{M}(0)=\mathcal{B}$.

Proof. 

• (i)
  Let $F_1,F_2\in \overline{M}(s),F_1\subset F_2$, then, $\forall n\in \mathbb{N}$ (where $\mathbb{N}=\{1,2,3,\dots \}$),

  $$\overline{\mu }(F_1)=\underset{n\to \infty }{lim\; sup}\frac{\#(F_1\cap \{0,1,\dots ,n-1\})}{n}\le \underset{n\to \infty }{lim\; sup}\frac{\#(F_2\cap \{0,1,\dots ,n-1\})}{n}=\overline{\mu }(F_2)$$

  Thus, $F_1\in \overline{M}(s)$ (i.e., $\overline{\mu }(F_1)\ge s$) implies $F_2\in \overline{M}(s)$ (i.e., $\overline{\mu }(F_1)\ge s$). So, $\overline{M}(s)(\forall s\in [0,1])$ are Furstenberg families.

• (ii)
  Let $F\in \overline{M}(s)$, that is, $\overline{\mu }(F)=\underset{n\to \infty }{lim\; sup}\frac{\#(F\cap \{0,1,\dots ,n-1\})}{n}\ge s$. Denote $F=\{t_1,t_2,\cdots \}$ (where $t_k\in {\mathbb{Z}}^{+}$, $t_{k_1}<t_{k_2}(k_1<k_2)$), then $F+i=\{t_1+i,t_2+i,\cdots \}$ and $F-i=\{t_{k_1}-i,t_{k_2}-i,\cdots \}(t_{k_j}-i\ge 0)$ for any $i\in {\mathbb{Z}}^{+}$.

  $$\underset{n\to \infty }{lim\; sup}\frac{\#((F+i)\cap \{0,1,\dots ,n-1\})}{n}=\underset{n\to \infty }{lim\; sup}\frac{\#(\{t_1+i,t_2+i,\cdots \}\cap \{0,1,\dots ,n-1\})}{n}$$

  $$=\underset{n\to \infty }{lim\; sup}\frac{\#(\{t_1,t_2,\cdots \}\cap \{0,1,\dots ,n-1\})}{n}=\overline{\mu }(F)\ge s$$

  and

  $$\underset{n\to \infty }{lim\; sup}\frac{\#((F-i)\cap \{0,1,\dots ,n-1\})}{n}\ge \underset{n\to \infty }{lim\; sup}\frac{\#(F\cap \{0,1,\dots ,n-1\})-i}{n}=\overline{\mu }(F)\ge s$$

  So, $\overline{M}(s)$ is shift-invariant.

• (iii)
  Obviously,

  $$\overline{M}(0)=\{F\in \mathcal{B}:\overline{\mu }(F)\ge 0\}=\{F\in \mathcal{B}:\underset{n\to \infty }{lim\; sup}\frac{\#(F\cap \{0,1,\dots ,n-1\})}{n}\ge 0\}=\mathcal{B}.$$

  This completes the proof.

☐

3. Properties $\mathit{P}(\mathit{k})$, $\mathit{Q}(\mathit{k})$ of Furstenberg Families

Definition 1.

Let k be a positive integer and $\mathcal{F}$ be a Furstenberg family.

• (1) For any $F\in \mathcal{F}$, if there exists an integer $j\in \{0,1,\cdots ,k-1\}$ such that $F_{k,j}=\{i\in {\mathbb{Z}}^{+}:ki+j\in F\}\in \mathcal{F}$, we say $\mathcal{F}$ have property $P(k)$;
• (2) If $F_k=\{ki+j\in {\mathbb{Z}}^{+}:j\in \{0,1,\cdots ,k-1\},i\in F\}\in \mathcal{F}$, we say $\mathcal{F}$ have property $Q(k)$.

The following proposition is given by [24]. For completeness, we give the proofs.

Proposition 2.

For any $s\in [0,1]$ and any $k\in {\mathbb{Z}}^{+}$, $\overline{M}(s)$ have properties $P(k)$ and $Q(k)$.

Proof. 

• (1)If $k=1$, $\forall F\in \overline{M}(s)$, $F_{1,0}=\{i\in {\mathbb{Z}}^{+}:i\in F\}=F$, i.e., there exists an integer $j=0$ such that $F_{k,j}\in \overline{M}(s)$. The following will discuss the case $k>1$.

  If $s=0$, $\overline{M}(0)=\mathcal{B}$. $\forall F\in \mathcal{B}$, $\forall k\in {\mathbb{Z}}^{+}$, obviously, there exist $j\in \{0,1,\dots ,k-1\}$ such that $F_{k,j}\in \mathcal{B}$.

  If $0<s\le 1$, suppose properties $P(k)$ does not hold. Then there exists a $F\in \overline{M}(s)$ such that $\overline{\mu }(F_{k,j})<s$ for every $j\in \{0,1,\dots ,k-1\}$.

  For any $j\in \{0,1,\dots ,k-1\}$, put ${\epsilon }_j>0$ which satisfied $\overline{\mu }(F_{k,j})<s-{\epsilon }_j$. One can find a sufficiently large number N such that, $n\ge N$, ${\#}_n(F_{k,j})<n(s-{\epsilon }_j)$ (where ${\#}_n(F_{k,j})$ denotes the cardinality of the set $F_{k,j}\cap \{0,1,\dots ,n-1\}$). Then ${\#}_n(F_{k,j}^c)>n-n(s-{\epsilon }_j)$, where $F_{k,j}^c$ denotes the complementary set of $F_{k,j}$.

  Give an integer $m=kn+l_m>kN$, $l_m\in \{0,1,\dots ,k-1\}$. By the definition of $F_{k,j}$, $ki+j\notin F$ if $i\notin F_{k,j}$. And $k{i}_1+j_1\ne k{i}_2+j_2$ if $i_1,i_2\in \{0,1,\dots ,n-1\}$, $j_1,j_2\in \{0,1,\dots ,k-1\}$ and $j_1\ne j_2$. Then

  $${\#}_m(F^c)\ge \sum _{j=0}^{k-1}{\#}_n(F_{k,j}^c)>\sum _{j=0}^{k-1}(n-n(s-{\epsilon }_j)).$$
  So,

  $${\#}_m(F)<m-\sum _{j=0}^{k-1}(n-n(s-{\epsilon }_j)).$$

  Put $\epsilon =min\{{\epsilon }_j:j=0,1,\dots ,k-1\}$, then

  $$\overline{\mu }(F)=\underset{n\to \infty }{lim\; sup}\frac{{\#}_m(F)}{m}\le \underset{n\to \infty }{lim}\frac{m-{\sum }_{j=0}^{k-1}(n-n(s-{\epsilon }_j))}{m}\le \underset{n\to \infty }{lim}\frac{m-k(n-n(s-\epsilon ))}{m}$$

  $$=\underset{n\to \infty }{lim}\frac{kn+l_m-kn+kn(s-\epsilon )}{kn+l_m}=s-\epsilon <s$$

  This contradicts to $\overline{\mu }(F)\ge s$.

• (2)Similarly, just consider the case $k>1$, $0<s\le 1$.
  Suppose properties $Q(k)$ does not hold. Then there exists an integer $F\in \overline{M}(s)$ such that $\overline{\mu }(F_k)<s$. Put $\epsilon >0$ which satisfied $\overline{\mu }(F_k)<s-\epsilon$. One can find a sufficiently large number N such that, $m\ge N$, ${\#}_m(F_k)<m(s-\epsilon )$. Give a $m=kn+l_m>kN(m\ge N)$, $l_m\in \{0,1,\dots ,k-1\}$. By the definition of $F_k$, $ki+j\in F_k(j\in \{0,1,\dots ,k-1\})$ if $i\in F$. And $k{i}_1+j_1\ne k{i}_2+j_2$ if $i_1\ne i_2$ and $j_1,j_2\in \{0,1,\dots ,k-1\}$. Then

  $$k({\#}_n(F))\le {\#}_m(F_k)<m(s-\epsilon ).$$
  So,

  $$\overline{\mu }(F)\le \underset{n\to \infty }{lim}\frac{m(s-\epsilon )}{kn}=\underset{n\to \infty }{lim}\frac{(kn+l_m)(s-\epsilon )}{kn}=s-\epsilon \le s.$$

  This contradicts to $\overline{\mu }(F)\ge s$.

  This completes the proof.

☐

4. $({\mathcal{F}}_{\mathbf{1}},{\mathcal{F}}_{\mathbf{2}})$-Chaos in Non-Autonomous Systems

Now, we state the definition of $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos in nonautonomous systems.

Definition 2.

Let $(X,\rho )$ be a compact metric space, ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ are two Furstenberg families. $\mathcal{D}\subset X$ is called a $({\mathcal{F}}_1,{\mathcal{F}}_2)$-scrambled set of $(X,f_{1,\infty })$ (briefly, $f_{1,\infty }$), if $\forall x\ne y\in \mathcal{D}$, the following two conditions are satisfied:

• (i) $\forall t>0$, $\left\{n\in \mathbb{N}:\rho (f_1^n(x),f_1^n(y))<t\right\}\in {\mathcal{F}}_1$;
• (ii) $\exists \delta >0$, $\left\{n\in \mathbb{N}:\rho (f_1^n(x),f_1^n(y))>\delta \right\}\in {\mathcal{F}}_2$.

The pair $(x,y)$ which satisfies the above two conditions is called an $({\mathcal{F}}_1,{\mathcal{F}}_2)$-scrambled pair of $f_{1,\infty }$.

$f_{1,\infty }$ is said to be $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic if there exists an uncountable $({\mathcal{F}}_1,{\mathcal{F}}_2)$-scrambled set of $f_{1,\infty }$. If ${\mathcal{F}}_1={\mathcal{F}}_2=\mathcal{F}$, $f_{1,\infty }$ is said to be $\mathcal{F}$-chaotic and $(x,y)$ is an $\mathcal{F}$-scrambled pair. $f_{1,\infty }$ is said to be strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic if there are some $\delta >0$ and an uncountable subset $\mathcal{D}\subset X$ such that for any $x,y\in \mathcal{D}$ with $x\ne y$, the following two conditions holds:

• (i) $\left\{n\in \mathbb{N}:\rho (f_1^n(x),f_1^n(y))<t\right\}\in {\mathcal{F}}_1$ for all $t>0$;
• (ii) $\left\{n\in \mathbb{N}:\rho (f_1^n(x),f_1^n(y))>\delta \right\}\in {\mathcal{F}}_2$.

$f_{1,\infty }$ is said to be strong $\mathcal{F}$-chaos if it is strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic and ${\mathcal{F}}_1={\mathcal{F}}_2=\mathcal{F}$.

Let us recall the definitions of Li-Yorke chaos and distributional chaos in non-autonomous systems (see [25,26]).

Definition 3.

Assume that $(X,f_{1,\infty })$ is a non-autonomous discrete system. If $x,y\in X$ with $x\ne y$, $(x,y)$ is called a Li–Yorke pair if

$$\underset{n\to \infty }{lim\; sup}\rho (f_1^n(x),f_1^n(y))>0and\underset{n\to \infty }{lim\; inf}\rho (f_1^n(x),f_1^n(y))=0.$$

The set $\mathcal{D}\subset X$ is called a Li–Yorke scrambled set if all points $x,y\in \mathcal{D}$ with $x\ne y$, $(x,y)$ is a Li–Yorke pair. $f_{1,\infty }$ is Li–Yorke chaotic if X contains an uncountable Li–Yorke scrambled set.

Assume that $(X,f_{1,\infty })$ is a non-autonomous discrete system. For any pair of points $x,y\in X$, define the upper and lower (distance) distributional functions generated by $f_{1,\infty }$ as

$$F_{xy}^{*}(t,f_{1,\infty })=\underset{n\to \infty }{lim\; sup}\frac{1}{n}\sum _{i=1}^n{\chi }_{[0,t)}(\rho (f_1^i(x),f_1^i(y)))$$

and

$$F_{xy}(t,f_{1,\infty })=\underset{n\to \infty }{lim\; inf}\frac{1}{n}\sum _{i=1}^n{\chi }_{[0,\delta )}(\rho (f_1^i(x),f_1^i(y)))$$

respectively. Where ${\chi }_{[0,t)}$ is the characteristic function of the set $[0,t)$, i.e., ${\chi }_{[0,t)}(a)=1$ when $a\in [0,t)$ or ${\chi }_{[0,t)}(a)=0$ when $a\notin [0,t)$.

Definition 4.

$f_{1,\infty }$ is distributionally chaotic if exists an uncountable subset $D\subset X$ such that for any pair of distinct points $x,y\in D$, we have that $F_{xy}^{*}(t,f_{1,\infty })=1$ for all $t>0$ and $F_{xy}(t,f_{1,\infty })=0$ for some $\delta >0$.

The set D is a distributionally scrambled set and the pair $(x,y)$ a distributionally chaotic pair.

It is not difficult to obtain that the pair $(x,y)$ is a $(\overline{M}(0),\overline{M}(0))$-scrambled pair if and only if $(x,y)$ is a Li–Yorke scrambled pair, and the pair $(x,y)$ is a $(\overline{M}(1),\overline{M}(1))$-scrambled pair if and only if $(x,y)$ is a distributionally scrambled pair. In fact,

$$\overline{M}(0)=\mathcal{B},\overline{M}(1)=\{F\in \mathcal{B}:\underset{n\to \infty }{lim\; sup}\frac{\#(F\cap \{1,2,\dots ,n\})}{n}=1\}.$$

Then, $\{n\in \mathbb{N}:\rho (f_1^n(x),f_1^n(y))<t\}\in \overline{M}(0)$ for any $t>0$ and $\{n\in \mathbb{N}:\rho (f_1^n(x),f_1^n(y))>\delta \}\in \overline{M}(0)$ for some $\delta >0$ is equivalent to that $\underset{n\to \infty }{lim\; sup}\rho (f_1^n(x),f_1^n(y))>0$ and $\underset{n\to \infty }{lim\; inf}\rho (f_1^n(x),f_1^n(y))=0$. $\{n\in \mathbb{N}:\rho (f_1^n(x),f_1^n(y))<t\}\in \overline{M}(1)$ for any $t>0$ and $\{n\in \mathbb{N}:\rho (f_1^n(x),f_1^n(y))>\delta \}\in \overline{M}(1)$ for some $\delta >0$ is equivalent to that $F_{xy}^{*}(t,f_{1,\infty })=1$ and $F_{xy}(\delta ,f_{1,\infty })=0$.

Hence, $(\overline{M}(0),\overline{M}(0))$-chaos is Li–Yorke chaos and $(\overline{M}(1),\overline{M}(1))$-chaos is distributional Chaos.

5. Main Results

Theorem 1.

Let ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ are two Furstenberg families with property $P(k)$, where k is a positive integer. ${\mathcal{F}}_1$ is positive shift-invariant. If the system $(X,f_{1,\infty })$ is $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos, then the system $(X,f_{1,\infty }^{\left[k\right]})$ is $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos too.

Proof. 

If D is an $({\mathcal{F}}_1,{\mathcal{F}}_2)$-scrambled set of $f_{1,\infty }$, the following proves that D is an $({\mathcal{F}}_1,{\mathcal{F}}_2)$-scrambled set of $f_{1,\infty }^{\left[k\right]}$.

• (i)Since X is compact and $f_i(i\in \mathbb{N})$ are continuous, then, for any $j\in \{1,2,\dots ,k-1\}$, $f_{s_1},\dots ,f_{s_{k-j}}$ are uniformly continuous (where $f_{s_1},\dots ,f_{s_{k-j}}$ are freely chosen from the sequence $f_i(i\in \mathbb{N})$). That is, for any $\delta >0$, there exists a ${\delta }^{*}>0$, $\forall a,b\in X$, $\rho (a,b)<{\delta }^{*}$ implies $\rho (f_{s_{k-j}}\circ \cdots \circ f_{s_1}(a),f_{s_{k-j}}\circ \cdots \circ f_{s_1}(b))<\delta$ ($j=1,2,\dots ,k-1$).
  Since D is an $({\mathcal{F}}_1,{\mathcal{F}}_2)$-scrambled set of $f_{1,\infty }$, then, $\forall x\ne y\in D$, for the above ${\delta }^{*}$, we have

  $$F=\{n\in \mathbb{N}:\rho (f_1^n(x),f_1^n(y))<{\delta }^{*}\}\in {\mathcal{F}}_1.$$
  And because ${\mathcal{F}}_1$ have property $P(k)$, there exists some $j\in \{1,2,\dots ,k-1\}$ such that

  $$F_{k,j}=\{i\in {\mathbb{Z}}^{+}:ki+j\in F\}=\{i\in {\mathbb{Z}}^{+}:\rho (f_1^{ki+j}(x),f_1^{ki+j}(y))<{\delta }^{*}\}\in {\mathcal{F}}_1.$$
  By the selection of ${\delta }^{*}$, we put $s_r=ki+j+r(r=1,2,\dots ,k-j)$, then

  $$F_{k,j}\subset \{i\in {\mathbb{Z}}^{+}:\rho (f_1^{ki+j+k-j}(x),f_1^{ki+j+k-j}(y))<\delta \}=\{i\in {\mathbb{Z}}^{+}:\rho (f_1^{k(i+1)}(x),f_1^{k(i+1)}(y))<\delta \}.$$

  Write $F_{k,j}+1=\{i+1:i\in {\mathbb{Z}}^{+},ki+j\in {\mathcal{F}}_1\}(\forall j=1,2,\dots ,k-1)$, then $F_{k,j}+1\subset \{i\in {\mathbb{Z}}^{+}:\rho (f_1^{ki}(x),f_1^{ki}(y))<\delta \}$.

  By the positive shift-invariant of ${\mathcal{F}}_1$ and $F_{k,j}\in {\mathcal{F}}_1$, we have $F_{k,j}+1\in {\mathcal{F}}_1$. And with the hereditary upwards of ${\mathcal{F}}_1$, for any $x,y\in D:x\ne y$, $\forall \delta >0$, $\{i\in {\mathbb{Z}}^{+}:\rho (f_1^{ki}(x),f_1^{ki}(y))<\delta \}\in {\mathcal{F}}_1$.

• (ii)
  Since D is a $({\mathcal{F}}_1,{\mathcal{F}}_2)$-scrambled set of $f_{1,\infty }$, then, for the above $x,y\in D(x\ne y)$, $\exists {\epsilon }^{*}>0$, such that $E=\{n\in {\mathbb{Z}}^{+}:\rho (f_1^n(x),f_1^n(y))>{\epsilon }^{*}\}\in {\mathcal{F}}_2$. And because ${\mathcal{F}}_2$ have property $P(k)$, then, there exists some $j\in \{1,2,\dots ,k-1\}$ such that

  $$E_{k,j}=\{i\in {\mathbb{Z}}^{+}:ki+j\in E\}=\{i\in {\mathbb{Z}}^{+}:\rho (f_1^{ki+j}(x),f_1^{ki+j}(y))>{\epsilon }^{*}\}\in {\mathcal{F}}_2.$$

  X is compact and $f_i(i\in \mathbb{N})$ are continuous, then, for any $j\in \{1,2,\dots ,k-1\}$, $f_{s_1},\dots ,f_{s_j}$ are uniformly continuous (where $f_{s_1},\dots ,f_{s_j}$ are freely chosen from the sequence $f_i(i\in \mathbb{N})$). For the above ${\epsilon }^{*}>0$, $\exists \epsilon >0$, $\forall p,q\in X$ satisfied $\rho (p,q)\le \epsilon$, inequality $\rho (f_{s_j}\circ \cdots \circ f_{s_1}(p),f_{s_j}\circ \cdots \circ f_{s_1}(q))\le {\epsilon }^{*}$ holds.

  The following will prove that $\{i\in {\mathbb{Z}}^{+}:\rho (f_1^{ki}(x),f_1^{ki}(y))>\epsilon \}\in {\mathcal{F}}_2$.

  Suppose $\{i\in {\mathbb{Z}}^{+}:\rho (f_1^{ki}(x),f_1^{ki}(y))>\epsilon \}\notin {\mathcal{F}}_2$, then

  $${\mathbb{Z}}^{+}-\{i\in {\mathbb{Z}}^{+}:\rho (f_1^{ki}(x),f_1^{ki}(y))>\epsilon \}=\{i\in {\mathbb{Z}}^{+}:\rho (f_1^{ki}(x),f_1^{ki}(y))\le \epsilon \}\in k{\mathcal{F}}_2.$$
  By the selection of ${\epsilon }^{*}$, we put $s_r=ki+r(r=1,2,\dots ,j)$, then

  $$\{i\in {\mathbb{Z}}^{+}:\rho (f_1^{ki+j}(x),f_1^{ki+j}(y))\le {\epsilon }^{*}\}\in k{\mathcal{F}}_2.$$
  So,

  $$\{i\in {\mathbb{Z}}^{+}:\rho (f_1^{ki+j}(x),f_1^{ki+j}(y))>{\epsilon }^{*}\}\notin k{\mathcal{F}}_2,$$

  This contradicts $E_{k,j}\in {\mathcal{F}}_2$.

  Hence, for $x\ne y\in D$ in (i), there exists a $\epsilon >0$ such that $\{i\in {\mathbb{Z}}^{+}:\rho (f_1^{ki}(x),f_1^{ki}(y))>\epsilon \}\in {\mathcal{F}}_2$.

  Combining with (i) and (ii), $f_{1,\infty }^{\left[k\right]}$ is $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos.

  This completes the proof.

☐

Theorem 2.

Let ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ are two Furstenberg families with property $Q(k)$, where k is a positive integer. ${\mathcal{F}}_2$ is negative shift-invariant. If the system $(X,f_{1,\infty }^{\left[k\right]})$ is $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos, then the system $(X,f_{1,\infty })$ is $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos too.

Proof. 

If D is a $({\mathcal{F}}_1,{\mathcal{F}}_2)$-scrambled set of $f_{1,\infty }^{\left[k\right]}$, the following prove that D is a $({\mathcal{F}}_1,{\mathcal{F}}_2)$-scrambled set of $f_{1,\infty }$.

• (i)Similar to Theorem 1, for any $j\in \{1,2,\dots ,k-1\}$, $f_{s_1},\dots ,f_{s_j}$ are uniformly continuous (where $f_{s_1},\dots ,f_{s_j}$ are freely chosen from the sequence $f_i(i\in \mathbb{N})$). That is, for any $\delta >0$, there exists a ${\delta }^{*}>0$, $\forall a,b\in X$, $\rho (a,b)<{\delta }^{*}$ implies $\rho (f_{s_j}\circ \cdots \circ f_{s_1}(a),f_{s_j}\circ \cdots \circ f_{s_1}(b))<\delta$ ($j=1,2,\dots ,k-1$).
  For any pair of distinct points $x,y\in D$, for the above ${\delta }^{*}$, one has

  $$F=\{n\in {\mathbb{Z}}^{+}:\rho (f_1^{kn}(x),f_1^{kn}(y))<{\delta }^{*}\}\in {\mathcal{F}}_1.$$
  By the selection of ${\delta }^{*}$, for $\forall n\in F$, $\forall j\in \{1,2,\cdots ,k-1\}$, put $s_r=ki+j+r(r=1,2,\dots ,j)$, then $\rho (f_1^{kn+j}(x),f_1^{kn+j}(y))<\delta$. And because ${\mathcal{F}}_1$ have property $Q(k)$, then

  $$F_k=\{kn+j\in {\mathbb{Z}}^{+}:j=1,2,\dots ,k-1,n\in F\}\in {\mathcal{F}}_1.$$

  Notice that $F_k\subset \{m\in {\mathbb{Z}}^{+}:\rho (f_1^m(x),f_1^m(y))<\delta \}$, then $\{m\in {\mathbb{Z}}^{+}:\rho (f_1^m(x),f_1^m(y))<\delta \}\in {\mathcal{F}}_1$.

• (ii)Since D is an $({\mathcal{F}}_1,{\mathcal{F}}_2)$-scrambled set of $f_{1,\infty }^{\left[k\right]}$, then, for the above $x,y\in D(x\ne y)$, there exist ${\epsilon }^{*}>0$, such that $E=\{n\in {\mathbb{Z}}^{+}:\rho (f_1^{kn}(x),f_1^{kn}(y))>{\epsilon }^{*}\}\in {\mathcal{F}}_2$.

  For any $j\in \{1,2,\dots ,k-1\}$, $f_{s_1},\dots ,f_{s_j}$ are uniformly continuous (where $f_{s_1},\dots ,f_{s_j}$ are freely chosen from the sequence $f_i(i\in \mathbb{N})$), then, for the above ${\epsilon }^{*}>0$, there exist $\epsilon >0$ such that $\rho (p,q)<\epsilon (p,q\in X)$ implies $\rho (f_{s_j}\circ \cdots \circ f_{s_1}(p),f_{s_j}\circ \cdots \circ f_{s_1}(q))\le {\epsilon }^{*}(j=1,2,\dots ,k-1)$. That is, $\rho (f_1^k(p),f_1^k(q))>{\epsilon }^{*}(p,q\in X)$ implies $\rho (f_1^j(p),f_1^j(q))>\epsilon (j=1,2,\dots ,k-1)$.

  $\forall n\in E$, $\forall j=1,2,\dots ,k-1$, put $s_r=k(n-1)+r(r=1,2,\dots ,j)$, then

  $$\rho (f_1^{k(n-1)+j}(x),f_1^{k(n-1)+j}(y))>\epsilon .$$

  Since ${\mathcal{F}}_2$ is negative shift-invariant, then $E-1\in {\mathcal{F}}_2$. And because ${\mathcal{F}}_2$ have property $Q(k)$, then ${(E-1)}_k\in {\mathcal{F}}_2$, i.e., $\{k(n-1)+j\in {\mathbb{Z}}^{+}:n-1\in E-1,j=1,2,\dots ,k-1\}\in {\mathcal{F}}_2$. Combining ${(E-1)}_k\subset \{m\in {\mathbb{Z}}^{+}:\rho (f_1^m(x),f_1^m(y))>\epsilon \}$ with the hereditary upwards of ${\mathcal{F}}_2$, we have $\{m\in {\mathbb{Z}}^{+}:\rho (f_1^m(x),f_1^m(y))>\epsilon \}\in {\mathcal{F}}_2$.

  By (i) and (ii), D is an $({\mathcal{F}}_1,{\mathcal{F}}_2)$-scrambled set of $f_{1,\infty }$.

  This completes the proof.

☐

Similarly, the following corollaries hold.

Corollary 1.

Let ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ are two Furstenberg families with property $P(k)$, where k is a positive integer. ${\mathcal{F}}_1$ is positive shift-invariant. If the system $(X,f_{1,\infty })$ is $\mathcal{F}$-chaos (strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos, or strong $\mathcal{F}$-chaos), then the system $(X,f_{1,\infty }^{\left[k\right]})$ is $\mathcal{F}$-chaos (strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos, or strong $\mathcal{F}$-chaos).

Corollary 2.

Let ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ are two Furstenberg families with property $Q(k)$, where k is a positive integer. ${\mathcal{F}}_2$ is negative shift-invariant. If the system $(X,f_{1,\infty }^{\left[k\right]})$ is $\mathcal{F}$-chaos (strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos, or strong $\mathcal{F}$-chaos), then the system $(X,f_{1,\infty })$ is $\mathcal{F}$-chaos (strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos, or strong $\mathcal{F}$-chaos).

Combining with Propositions 1 and 2, Theorems 1 and 2, and Corollarys 1 and 2, the following conclusions are obtained.

Theorem 3.

Let s and t are arbitrary two numbers in $[0,1]$, then

• (1) If D is an $(\overline{M}(s),\overline{M}(t))$-scrambled set (or strong $(\overline{M}(s),\overline{M}(t))$-scrambled set) of $f_{1,\infty }$, then, for every $k\in {\mathbb{Z}}^{+}$, D is an $(\overline{M}(s),\overline{M}(t))$-scrambled set(or strong $(\overline{M}(s),\overline{M}(t))$-scrambled set) of $f_{1,\infty }^{\left[k\right]}$.
• (2) For some positive integer k, if D is an $(\overline{M}(s),\overline{M}(t))$-scrambled set (or strong $(\overline{M}(s),\overline{M}(t))$-scrambled set) of $f_{1,\infty }^{\left[k\right]}$, then D is an $(\overline{M}(s),\overline{M}(t))$-scrambled set (or strong $(\overline{M}(s),\overline{M}(t))$-scrambled set) of $f_{1,\infty }$.

Proof. 

• (1)By Proposition 1, $\overline{M}(s)$ is shift-invariant (obviously positive shift-invariant). And because $\overline{M}(s),\overline{M}(t)$ are two Furstenberg families with property $P(k)$ (Proposition 2). Then, according to the proof of Theorem 1, if D is an $(\overline{M}(s),\overline{M}(t))$-scrambled set of $f_{1,\infty }$, then, for every $k\in {\mathbb{Z}}^{+}$, D is an $(\overline{M}(s),\overline{M}(t))$-scrambled set of $f_{1,\infty }^{\left[k\right]}$.
• (2)In the same way, (2) holds.

  This completes the proof.

☐

With the preparations in Section 4, we have

Corollary 3.

• (1) If D is a Li–Yorke scrambled set (or distributionally scrambled set) of $f_{1,\infty }$, then, for every $k\in {\mathbb{Z}}^{+}$, D is a Li–Yorke scrambled set (or distributionally scrambled set) of $f_{1,\infty }^{\left[k\right]}$.
• (2) For some positive integer k, if D is a Li–Yorke scrambled set (or distributionally scrambled set) of $f_{1,\infty }^{\left[k\right]}$, then, D is a Li–Yorke scrambled set (or distributionally scrambled set) of $f_{1,\infty }$.

Remark 1.

In the non-autonomous systems, the iterative properties of Li–Yorke chaos and distributional chaos are discussed in [25,26] before. The conclusions in Corollary 3 remains consistent with them.

This paper has presented several properties of $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos, strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos, and strong $\mathcal{F}$-chaos. There are some other problems, such as generically $\mathcal{F}$-chaos and $\mathcal{F}$-sensitivity, to discuss. Moreover, property $P(k)$ is closely related to congruence theory. Follow this line, one can consider other Furstenberg families which consist of number sets with some special characteristics.

Author Contributions

Tianxiu Lu proposed the idea; Most of the conclusions are proved by Xiao Tang and Tianxiu Lu; Some important steps in Theorem 1 and Theorem 2 are inferred by Guanrong Chen; Tianxiu Lu wrote the paper.

Conflicts of Interest

The authors declare no conflict of interest.