Collective Sensitivity, Collective Accessibility, and Collective Kato’s Chaos in Duopoly Games

Abstract

By using the uniform continuity of two onto maps, this paper further explores stronger forms of Kato’s chaos, sensitivity, and accessibility of Cournot maps. In particular, the sensitivity, the collective sensitivity, the accessibility, and the collective accessibility of the compositions of two reaction functions are studied. It is observed that a Cournot onto map H on a product space is sensitive (collectively sensitive, collectively accessible, accessible, or collectively Kato chaotic) if and only if the restriction of the map $H^2$ to the MPE-set is sensitive as well. Several examples are given to show the necessity of the reaction functions being continuous onto maps.

1. Introduction

Let $\mathbb{E},\mathbb{F}\subset \mathbb{R}$ be closed intervals. $h_1:\mathbb{F}\to \mathbb{E}$ and $h_2:\mathbb{E}\to \mathbb{F}$ be continuous onto maps, and let $H:\mathbb{E}\times \mathbb{F}\to \mathbb{E}\times \mathbb{F}$ be defined by $H(a,b)=(h_1\left(b\right),h_2\left(a\right))$ for any $(a,b)\in \mathbb{E}\times \mathbb{F}$. Such a map H has been posed to present a mathematical description of competition in a duopolistic market, which is called Cournot duopoly [1]. Then, the map H said to be a Cournot map, and $h_1$ and $h_2$ are said to be reaction functions which give laws to organize the production of some firms, such that they are competitors in the market. Consequently, these games which are given by Cournot maps are called duopoly games. The first definition of chaos in a mathematically rigorous way was probably provided by Li and Yorke [2]. Since then, a lot of different definitions of chaos have been presented. Each of them is used to reflect some kind of unpredictability of a given system in its evolution. Akin and Kolyada [3] gave the definitions of Li–Yorke sensitivity and spatiotemporal chaos for the first time. Schweizer and Smítal [4] gave the definition of distributional chaos. It is well known that distributional chaos is equivalent to positive topological entropy and some other definitions of chaos for some kinds of spaces [4,5], and that this equivalence relationship does not hold for higher-dimensional spaces [6] and zero-dimensional spaces [7]. In [8], Wang et al. defined distributional chaos with respect to a sequence and proved that it is equivalent to Li–Yorke chaos for continuous maps over a given closed interval. Over the past few decades, many scholars have paid very close attention to the dynamic behavior of Cournot maps [1,9,10,11,12,13]. It is known from [1,12] that there are Markov perfect equilibria processes. Concretely speaking, two fixed players move alternatively, such that each of them chooses the best reply to the previous action of another player. Put ${\mathbb{M}}_1=\{(h_1\left(b\right),b):b\in \mathbb{F}\}$, ${\mathbb{M}}_2=\{(a,h_2\left(a\right)):a\in \mathbb{E}\}$ and ${\mathbb{M}}_{12}={\mathbb{M}}_1\cup {\mathbb{M}}_2$. It is clear that $H\left({\mathbb{M}}_{12}\right)\subset {\mathbb{M}}_{12}$. From [9], it is known that the set ${\mathbb{M}}_{12}$ is said to to be an MPE-set for H. Furthermore, in [9] the authors considered some definitions of chaos for Cournot maps, and proved that for each of the definitions they considered in [9], it is not satisfied that H is chaotic if and only if ${H|}_{{\mathbb{M}}_{12}}$ is also chaotic (i.e., the restriction of the map H to the set ${\mathbb{M}}_{12}$). Inspired chaotic properties of Cournot maps in [1,12,13,14,15,16], Lu and Zhu [17] further investigated the Li–Yorke chaos, distributional chaos, sensitivity, and Li–Yorke sensitivity of such maps. In particular, they showed that ${H|}_{{\mathbb{M}}_{12}}$, $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ enjoy some of the same chaotic properties. In [18], it was shown for any Cournot map $H(a,b)=(h_1\left(b\right),h_2\left(a\right))$ on the product space $\mathbb{E}\times \mathbb{F}$, the following statements are true.

(1) If H is Kato’s chaos, then at least one of $M^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ is Kato’s chaos.

(2) Suppose that $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ is Kato’s chaos. The maps $h_1$ and $h_2$ satisfy that, for any $\epsilon >0$, if

$$\mid {(h_2\circ h_1)}^n\left(b_1\right)-{(h_2\circ h_1)}^n\left(b_2\right)\mid <\epsilon$$

and

$$\mid {(h_1\circ h_2)}^m\left(a_1\right)-{(h_1\circ h_2)}^m\left(a_2\right)\mid <\epsilon$$

for some integers $n,m\ge 1$, then there exists an integer $l(n,m,\epsilon )>0$ that satisfies

$$\mid {(h_2\circ h_1)}^{l(n,m,\epsilon )}\left(b_1\right)-{(h_2\circ h_1)}^{l(n,m,\epsilon )}\left(b_2\right)\mid <\epsilon$$

and

$$\mid {(h_1\circ h_2)}^{l(n,m,\epsilon )}\left(a_1\right)-{(h_1\circ h_2)}^{l(n,m,\epsilon )}\left(a_2\right)\mid <\epsilon ,$$

and then H is Kato’s chaos. Here, $\mid ·\mid$ denotes 1-norm on $\mathbb{R}$, i.e. absolute value. In [19] Zhao and Li considered $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos (resp. strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos) for such maps and obtained the following results:

(1) H is $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic (resp. strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic) if and only if $H^2{|}_{{\mathbb{M}}_1}$ is $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic (resp. strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic) if and only if $H^2{|}_{{\mathbb{M}}_2}$ is $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic (resp. strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic).

(2) H is $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic (resp. strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic) if and only if $H^2{|}_{{\mathbb{M}}_1\cup {\mathbb{M}}_2}$ is $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic (resp. strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic).

(3) $h_1\circ h_2$ is $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic (resp. strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic) if and only if $h_2\circ h_1$ is as well.

In [20], Li and Lu discussed several chaotic properties of the s-dimensional dynamical system of the form $H(a_1,a_2,\cdots ,a_s)=(h_s\left(a_s\right),h_1\left(a_1\right),\cdots ,h_{s-1}\left(a_{s-1}\right))$, where $a_k\in {\mathbb{H}}_k$ for any $k\in \{1,2,\cdots ,s\}$, $s>1$ is a given integer, and ${\mathbb{H}}_k\subset \mathbb{R}=(-\infty ,+\infty )$ is a closed subinterval for any $k\in \{1,2,\cdots ,s\}$. In particular, a necessary and sufficient condition for a cyclic permutation map $H(a_1,a_2,\cdots ,a_s)=(h_s\left(a_s\right),h_1\left(a_1\right),\cdots ,h_{s-1}\left(a_{s-1}\right))$ to be LY-chaotic (i.e., chaotic in the sense of Li–Yorke) or h-chaotic (i.e., its topological entropy is positive) or RT-chaotic (i.e., chaotic in the sense of Ruelle–Takens) or D-chaotic (i.e., chaotic in the sense of Devaney) was established. Furthermore, the LY-chaoticity, h-chaoticity, RT-chaoticity, and D-chaoticity of such a cyclic permutation map were studied. Moreover, it was verified that the topological entropy $h\left(H\right)$ of such a cyclic permutation map is the same as the topological entropy of each of the following maps: $h_j\circ h_{j-1}\circ \cdots \circ h_1\circ h_s\circ h_{s-1}\circ \cdots \circ h_{j+1}$ if $j=1,\cdots ,s-1$ and $h_s\circ h_{s-1}\circ \cdots \circ h_1$, and that H is sensitive if and only if so is at least one of the coordinates maps of $H^s$. In [21], Linero Bas and Soler López introduced the notion of a cyclically permuted direct product map and considered the (totally) topological transitivity and the (weakly) topological mixing for a cyclically permuted direct product map by studying the relationship between the dynamics of F and that of the compositions $h_{\sigma \left(i\right)}\circ \cdots \circ h_{{\sigma }^m\left(j\right)}$, where $j\in \{1,\cdots ,m\}$, $F(a_1,a_2,\cdots ,a_m)=(h_{\sigma \left(1\right)}\left(a_{\sigma \left(1\right)}\right),h_{\sigma \left(2\right)}\left(a_{\sigma \left(2\right)}\right),\cdots ,h_{\sigma \left(m\right)}\left(a_{\sigma \left(m\right)}\right))$ said to be a cyclically permuted direct product map, which is defined from the Cartesian product ${\mathbb{X}}_1\times {\mathbb{X}}_2\times \cdots \times {\mathbb{X}}_m$ into itself (where ${\mathbb{X}}_1,{\mathbb{X}}_2,\cdots ,{\mathbb{X}}_m$ are general topological spaces, $h_{\sigma \left(i\right)}:{\mathbb{X}}_{\sigma \left(i\right)}\to {\mathbb{X}}_j$ is continuous for any $j\in \{1,\cdots ,m\}$, and $\sigma$ is a cyclic permutation of $\{1,2,\cdots ,m\},m\ge 2$). In [22], Linero Bas and Soler López got several results on transitivity for a cyclically permuted direct product map of the Cartesian product $J^m$, where $J=[0,1]$. In particular, it was verified that, for any integer $m>2$, the transitivity of this map F is equivalent to the total transitivity. If $m=2$, they deduced a splitting result for a transitive map. In addition, they extended well-known properties of transitivity from an interval map to a cyclically permuted direct product map. To do it, they used the strong link between the map F and the compositions ${\phi }_i=h_{\sigma \left(i\right)}\circ \cdots \circ h_{{\sigma }^m\left(i\right)},i=\{1,\cdots ,m\}$. From [20,21,22], it is known that, for cyclically permuted direct product maps, if $m=2$ and ${\mathbb{X}}_1={\mathbb{X}}_2=[0,1]$, they appear associated with Cournot duopoly.

It is known that the sensitive property characterizes the unpredictability of chaotic phenomenon. This kind of property is one of the essential conditions of various definitions of a chaotic system. Hence, when is a system sensitive? In [23], Moothathu initiated a preliminary study of stronger forms of sensitivity of continuous self-maps of compact metric spaces by using large subsets of N. Mainly he discussed syndetic sensitivity and cofinite sensitivity, and proved the following: (1) The syndetical transitivity of a non-minimal map implies its syndetical sensitivity. (2) The sensitivity of a map on the interval $[0,1]$ implies its cofinite sensitivity. (3) The sensitivity of a subshift of finite type implies its cofinite sensitivity. (4) The syndetical transitivity of an infinite subshift implies its syndetical sensitivity. (5) There is not a cofinitely sensitive Sturmian subshift. (6) There is a transitive, sensitive map which is not syndetically sensitive. In [24], Xu et al. observed that a mixing transformation of a manifold is sensitive and topologically transitive. In [25], He et al. obtained that for a measure-preserving map and a measure-preserving semi-flow on a metric probability space with a fully supported measure, if they are weak-mixing, then they are sensitive. In [26], M. Salman et al. defined sensitivity, multi-sensitivity, cofinite sensitivity, and syndetic sensitivity for non-autonomous dynamical systems on uniform spaces and got several sufficient conditions under which topological transitivity and dense periodic points imply sensitivity for non-autonomous systems on Hausdorff uniform spaces. Furthermore, they explored sensitivity and other stronger versions of sensitivity for the systems induced on hyperspaces and for the product of non-autonomous dynamical systems on uniform spaces. In [27], Vasisht and Ruchi Das researched a few stronger forms of transitivity in a non-autonomous discrete dynamical system $(\mathbb{E},h_{1,\infty })$, where $\left(h_{1,\infty }\right)$ is a sequence of continuous self-maps which converges uniformly to h. They gave notions of thick sensitivity, ergodic sensitivity and multi-sensitivity for a non-autonomous discrete dynamical system. These sensitivity properties are all stronger forms of sensitivity. Ref. [27] established that under certain conditions, if the rate of convergence at which $\left(h_j\right)$ converges to h is sufficiently fast, then various forms of sensitivity and transitivity for the above nonautonomous system are the same as the responding properties of the autonomous system $(\mathbb{E},h)$. Moreover, they presented counterexamples to support their results. In [28], Li et al. studied stronger forms of transitivity and sensitivity for a nonautonomous discrete dynamical system by using the Furstenberg family. In particular, they considered the $\mathcal{F}$-transitivity, $\mathcal{F}$-mixing, $\mathcal{F}$-sensitivity, $\mathcal{F}$-collective sensitivity, $\mathcal{F}$-synchronous sensitivity, (${\mathcal{F}}_1$,${\mathcal{F}}_2$)-sensitivity, and $\mathcal{F}$-multi-sensitivity for a nonautonomous discrete dynamical system and extended the responding results of [27]. In [29], F. Ghane et al. considered the equicontinuity and sensitivity of iterated function systems (for short, IFSs). In particular, they discussed more general case of IFSs (in other words, the IFSs which are generated by a family of relations). Ref. [29] generalized the notions of transitivity, sensitivity, and equicontinuity to these kinds of systems. The relationships between these notions are studied. Some sufficient conditions for sensitivity of IFSs are given. Then, they defined weak topologically exactness for IFSs that are generated by a family of relations. It was shown that non-minimal weak topologically exact IFSs are sensitive. This deduces different examples of nonminimal sensitive systems which are not an M-system. Moreover, some interesting examples are presented for providing some facts about the sensitive property of IFSs. In [30], Gu studied the relationships between Kato’s chaoticity of a dynamical system $(\mathbb{E},h)$ on a compact metric space $\mathbb{E}$ and Kato’s chaoticity of the set-valued discrete system $(\kappa \left(\mathbb{E}\right),\overline{h})$ which is associated with $(\mathbb{E},h)$. He proved that Kato’s chaoticity of $(\kappa \left(\mathbb{E}\right),\overline{h})$ implies the Kato’s chaoticity of $(\mathbb{E},h)$, and that $(\mathbb{E},h)$ is Kato chaotic and is equivalent to $(\kappa \left(\mathbb{E}\right),\overline{h})$ is Kato chaotic in $W^e$-topology. Moreover, if a continuous map with a fixed point from a complete metric space without isolated point into itself is Ruelle–Takens chaotic, then it is Kato chaotic. In [31], Wang et al. introduced and investigated strong Kato’s chaos for a group action on a compact metric space. For a compact metric space $\mathbb{E}$, which does not have isolated points and a topologically commutative group G over $\mathbb{E}$, if the weak mixing of the dynamical system $(\mathbb{E},G)$ deduces the chaoticity in the strong sense of Kato. In [32], Değirmenci and Koçak considered how chaos conditions on maps carry over to their products. First, they presented a counterexample which shows that the product of two Devaney chaotic maps need not be chaotic. They then showed that if two maps (or even one of them) is sensitive, then so is their product. In addition, they presented some sufficient conditions under which the product of two chaotic maps is Devaney chaotic.

Inspired by the above results, this paper will further discuss stronger forms of Kato’s chaos, sensitivity and accessibility of the chaotic properties of the above Cournot maps. The main purpose of this paper is to extend and improve the existing results for these maps by some stronger notions. In particular, it is proven that H is sensitive if and only if $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ are sensitive, where ${\mathbb{M}}_1=\{(h_1\left(c_2\right),c_2):c_2\in \mathbb{F}\}$, ${\mathbb{M}}_2=\{(c_1,h_2\left(c_1\right)):c_1\in \mathbb{E}\}$, H is collectively sensitive if and only if $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ are collectively sensitive. Additionally, a result, in [18] is extended to collective accessibility. For an onto Cournot map $H(p,r)=(h_1\left(r\right),h_2\left(p\right))$ on $\mathbb{E}\times \mathbb{F}$, the following two new results are established.

(1) $h_1\circ h_2$ is sensitive (resp. accessible) if and only if $h_2\circ h_1$ is as well.

(2) $h_1\circ h_2$ is collectively sensitive (resp. collectively accessible) if and only if $h_2\circ h_1$ is as well.

Moreover, this paper gives a few nontrivial examples which show that it is necessary to assume that $h_1:\mathbb{F}\to \mathbb{E}$ and $h_2:\mathbb{E}\to \mathbb{F}$ are continuous onto maps.

2. Preliminaries

Let $(\mathbb{E},d)$ be a metric space and $h:\mathbb{E}\to \mathbb{E}$ be a continuous map.

Definition 1.

A dynamic system $(\mathbb{E},h)$ (or the map $h:\mathbb{E}\to \mathbb{E}$) is

(1) topologically transitive if for any nonempty open subsets $U,V\subset \mathbb{E}$, $h^n\left(U\right)\cap V\ne \varnothing$ for some positive integer n;

(2) topologically mixing if for any nonempty open subsets $U,V\subset \mathbb{E}$, there is a positive integer n with $h^t\left(U\right)\cap V\ne \varnothing$ for any integer $t\ge n$;

(3) sensitive if there is a $\delta >0$, such that for any $\epsilon >0$ and any $a\in \mathbb{E}$, there exists a point $b\in \mathbb{E}$ with $d(a,b)<\epsilon$ and $d(h^n\left(a\right),h^n\left(b\right))>\delta$ for some integer $n\ge 1$, where δ is called a sensitivity constant of the map h;

(4) accessible if for any $\epsilon >0$ and any two nonempty open subsets $U,V\subset \mathbb{E}$, there are two points $a\in U$ and $b\in V$ with $d(h^n\left(a\right),h^n\left(b\right))<\epsilon$ for some integer $n\ge 1$; and

(5) chaotic in the sense of Ruelle and Takens if it is topologically transitive and sensitive.

Definition 2

([18]). Let $(\mathbb{E},h)$ be a dynamical system on a metric space $(\mathbb{E},d)$. The system $(\mathbb{E},h)$ is called to be accessible if for any $\delta >0$ and any nonempty open subsets $V_1,V_2\subset \mathbb{E}$, and there are $v_i\in V_i$ ($i\in \{1,2\}$) such that $d(h^n\left(v_1\right),h^n\left(v_2\right))<\delta$ for some integer $n>0$. The system $(\mathbb{E},h)$ or the map h is called to be Kato chaotic if h is sensitive and accessible.

It is well known that a topologically mixing dynamic system $(\mathbb{E},h)$ or a topologically mixing map $h:\mathbb{E}\to \mathbb{E}$ is Kato’s chaos [33].

Definition 3

([34]). Let $(\mathbb{E},h)$ be a dynamical system on a metric space $(\mathbb{E},d)$. The system $(\mathbb{E},h)$ is called to be collectively accessible, if for any $\delta >0$ and any nonempty open subsets $V_1^{\left(1\right)},V_2^{\left(1\right)},\cdots ,V_s^{\left(1\right)},V_1^{\left(2\right)},V_2^{\left(2\right)},\cdots ,V_t^{\left(2\right)}\subset \mathbb{E}$, there exist $v_i^{\left(1\right)}\in V_i^{\left(1\right)}$ for any $i\in \{1,2,\cdots ,s\}$, $v_j^{\left(2\right)}\in V_j^{\left(2\right)}$ for any $j\in \{1,2,\cdots ,t\}$ and some integer $n>0$ such that one of the following holds.

(1) There is an $i_0\in \{1,2,\cdots ,s\}$ such that $d(h^n\left(v_{i_0}^{\left(1\right)}\right),h^n\left(v_j^{\left(2\right)}\right))<\delta$ for any $j\in \{1,2,\cdots ,t\}$.

(2) There is a $j_0\in \{1,2,\cdots ,t\}$ such that $d(h^n\left(v_i^{\left(1\right)}\right),h^n\left(v_{j_0}^{\left(2\right)}\right))<\delta$ for any $i\in \{1,2,\cdots ,s\}$.

Clearly, collective accessibility implies accessibility.

Definition 4

([34]). Let $(\mathbb{E},h)$ be a dynamical system on a metric space $(\mathbb{E},d)$ and $\delta >0$ a constant. The system $(\mathbb{E},h)$ is called to be collectively sensitive with the collective sensitivity constant δ if for any finitely many distinct points $u_1,u_2,\cdots ,u_m\in \mathbb{E}$ and any $\epsilon >0$, there exist m distinct points $v_1,v_2,\cdots ,v_m\in \mathbb{E}$ such that the following two conditions are satisfied:

(1) $d(u_j,v_j)<\epsilon$ for all $1\le j\le m$;

(2) there exists a $j_0$ with $1\le j_0\le m$ such that for some integer $n>0$, $d(h^n\left(u_j\right),h^n\left(v_{j_0}\right))>\delta$ or $d(h^n\left(u_{j_0}\right),h^n\left(v_j\right))>\delta ,1\le j\le m$.

Clearly, collective sensitivity implies sensitivity.

Definition 5

([34]). Let $(\mathbb{E},h)$ be a dynamical system on a metric space $(\mathbb{E},d)$. The system $(\mathbb{E},h)$ is called to be collectively Kato-chaotic if it is collectively accessible and collectively sensitive.

Obviously, collective Kato’s chaos implies Kato’s chaos.

3. Main Results

Assume that $(\mathbb{E},d_1)$ and $(\mathbb{F},d_2)$ are compact metric spaces, and that $h_1:\mathbb{F}\to \mathbb{E}$ and $h_2:\mathbb{E}\to \mathbb{F}$ are uniformly continuous onto maps. The following Lemmas 1–4 are needed.

Lemma 1.

$h_1\circ h_2$ is sensitive if and only if so is $h_2\circ h_1$.

Proof. 

Assume that $h_1\circ h_2$ is sensitive with sensitivity constant $\beta$. Let $r\in \mathbb{F}$ and $U_{r,\epsilon }\subset \mathbb{F}$ be a neighbourhood of $r\in \mathbb{F}$. Because $h_2$ is continuous, one can choose $p\in h_2^{-1}\left(\left\{r\right\}\right)$, and $h_2^{-1}\left(U_{r,\epsilon }\right)$ is a neighbourhood of $p\in \mathbb{E}$. Since $h_1\circ h_2$ is sensitive with sensitivity constant $\beta$, there are $p^{\prime }\in h_2^{-1}\left(U_{r,\epsilon }\right)$ and an integer $j>0$, such that

$$d_1({(h_1\circ h_2)}^j\left(p\right),{(h_1\circ h_2)}^j\left(p^{\prime }\right))>\beta .$$

Because $h_2$ is uniformly continuous, then there is a ${\epsilon }^{\prime }>0$ such that, if $d_1(p,p^{\prime })\le \epsilon$ and $p,p^{\prime }\in \mathbb{E}$, then

$$d_2(h_2\left(p\right),h_2\left(p^{\prime }\right))\le {\epsilon }^{\prime }.$$

Hence,

$$d_2(h_2\circ {(h_1\circ h_2)}^j\left(p\right),h_2\circ {(h_1\circ h_2)}^j\left(p^{\prime }\right))>{\beta }^{\prime }$$

for some ${\beta }^{\prime }>0$. That is,

$$d_2({(h_2\circ h_1)}^j\left(r\right),{(h_2\circ h_1)}^j\left(r^{\prime }\right))>{\beta }^{\prime },$$

where $r^{\prime }=h_2\left(p^{\prime }\right)\in U_{r,\epsilon }$. By the definition, $h_2\circ h_1$ is sensitive with sensitivity constant ${\beta }^{\prime }$. □

Similarly, one can easily verify that if $h_2\circ h_1$ is sensitive, then $h_1\circ h_2$ is sensitive.

Remark 1.

For Lemma 1, it is necessary to assume that $h_1:\mathbb{F}\to \mathbb{E}$ and $h_2:\mathbb{E}\to \mathbb{F}$ are continuous onto maps.

Example 1.

Let $\mathbb{X}=[0,1]$ and $\mathbb{Y}=[0,2]$. $h_1:\mathbb{Y}\to \mathbb{X}$ is defined by $h_1\left(r\right)=1-|1-2r|$ for any $r\in \mathbb{Y}$ and $h_1\left(r\right)=0$ for any $r\in [1,2]$. $h_2:\mathbb{X}\to \mathbb{Y}$ is defined by $h_2\left(p\right)=1-|1-2p|$ for any $p\in \mathbb{X}$, where $|·|$ denotes absolute value. Then, the following results hold:

(1) $h_1\circ h_2$ is sensitive.

(2) $h_2\circ h_1$ is not sensitive.

Proof. 

Let $\lambda$ be a tent map on $[0,1]$ which is defined by $\lambda \left(x\right)=1-|1-2x|$ for any $x\in [0,1]$. It is known that $\lambda$ is topologically mixing. Then ${\lambda }^2$ is topologically mixing. Consequently, ${\lambda }^2$ is sensitive. By the definition of $h_1$ and $h_2$, one has $h_1\circ h_2={\lambda }^2$. This implies that $h_1\circ h_2$ is sensitive. It is easily seen that $h_2\circ h_1$ is not sensitive. □

Lemma 2.

$h_1\circ h_2$ is collectively sensitive if and only if so is $h_2\circ h_1$.

Proof. 

Assume that $h_1\circ h_2$ is collectively sensitive with collective sensitivity constant $\beta$. Let $r_1^{\left(1\right)},r_2^{\left(1\right)},\cdots ,r_m^{\left(1\right)}\in \mathbb{F}$ are m distinct points. Because $h_2$ is uniformly continuous, then one can choose $p_1^{\left(1\right)}\in h_2^{-1}\left(\left\{r_1^{\left(1\right)}\right\}\right),p_2^{\left(1\right)}\in h_2^{-1}\left(\left\{r_2^{\left(1\right)}\right\}\right),\cdots ,p_m^{\left(1\right)}\in h_2^{-1}\left(\left\{r_m^{\left(1\right)}\right\}\right)$ satisfying $p_i^{\left(1\right)}\ne p_j^{\left(1\right)}(i\ne j)$, and $h_2^{-1}\left(\left\{r_1^{\left(1\right)}\right\}\right),h_2^{-1}\left(\left\{r_2^{\left(1\right)}\right\}\right),\cdots ,h_2^{-1}\left(\left\{r_m^{\left(1\right)}\right\}\right)$ are neighbourhoods of $p_1^{\left(1\right)},p_2^{\left(1\right)},\cdots ,p_m^{\left(1\right)}$, respectively.

By the collective sensitivity of $h_1\circ h_2$, for any $\epsilon >0$, there exist m distinct points $p_1^{\left(2\right)},p_2^{\left(2\right)},\cdots ,p_m^{\left(2\right)}\in \mathbb{E}$, such that the following two conditions are satisfied.

(1) $d_1(p_j^{\left(1\right)},p_j^{\left(2\right)})<\epsilon$ for all $1\le j\le m$;

(2) there exists a $j_0:1\le j_0\le m$, such that for some integer $n_1>0$,

$$d_1({(h_1\circ h_2)}^{n_1}\left(p_j^{\left(1\right)}\right),{(h_1\circ h_2)}^{n_1}\left(p_{j_0}^{\left(2\right)}\right))>\beta$$

or

$$d_1({(h_1\circ h_2)}^{n_1}\left(p_{j_0}^{\left(1\right)}\right),{(h_1\circ h_2)}^{n_1}\left(p_j^{\left(2\right)}\right))>\beta ,1\le j\le m.$$

Let

$$r_1^{\left(2\right)}=h_2\left(p_1^{\left(2\right)}\right),r_2^{\left(2\right)}=h_2\left(p_2^{\left(2\right)}\right),\cdots ,r_m^{\left(2\right)}=h_2\left(p_m^{\left(2\right)}\right).$$

Because $h_2$ is uniformly continuous, then there exists a ${\epsilon }^{\prime }>0$ such that $d_2(r_j^{\left(1\right)},r_j^{\left(2\right)})<{\epsilon }^{\prime }$ for all $1\le j\le m$.

Furthermore, because $d_1({(h_1\circ h_2)}^{n_1}\left(p_j^{\left(1\right)}\right),{(h_1\circ h_2)}^{n_1}\left(p_{j_0}^{\left(2\right)}\right))>\beta$ (or $d_1({(h_1\circ h_2)}^{n_1}\left(p_{j_0}^{\left(1\right)}\right),$${(h_1\circ h_2)}^{n_1}\left(p_j^{\left(2\right)}\right))>\beta )$, then there exists a ${\beta }^{\prime }>0$ such that

$$d_2(h_1^{-1}{(h_1\circ h_2)}^{n_1}\left(p_j^{\left(1\right)}\right),h_1^{-1}{(h_1\circ h_2)}^{n_1}\left(p_{j_0}^{\left(2\right)}\right))>{\beta }^{\prime }$$

or

$$d_2(h_1^{-1}{(h_1\circ h_2)}^{n_1}\left(p_{j_0}^{\left(1\right)}\right),h_1^{-1}{(h_1\circ h_2)}^{n_1}\left(p_j^{\left(2\right)}\right))>{\beta }^{\prime }.$$

Otherwise, it contradicts that $h_1$ is a uniformly continuous map.

That is to say, there is an $n_1-1\in \mathbb{N}$, such that

$$d_2({(h_2\circ h_1)}^{n_1-1}\left(r_j^{\left(1\right)}\right),{(h_2\circ h_1)}^{n_1-1}\left(r_{j_0}^{\left(2\right)}\right))>{\beta }^{\prime }$$

or

$$d_2({(h_2\circ h_1)}^{n_1-1}\left(r_{j_0}^{\left(1\right)}\right),{(h_2\circ h_1)}^{n_1-1}\left(r_j^{\left(2\right)}\right))>{\beta }^{\prime }.$$

Thus, $h_2\circ h_1$ is collectively sensitive. □

Lemma 3.

$h_1\circ h_2$ is accessible if and only if so is $h_2\circ h_1$.

Proof. 

Assume that $h_1\circ h_2$ is accessible, and that ${\mathbb{V}}_j\subset \mathbb{F}$ is nonempty and open for each $j\in \{1,2\}$. Set $\lambda >0$. Because $h_2$ is uniformly continuous, there is a ${\lambda }^{\prime }>0$ such that if $d_1(p_1,p_2)<{\lambda }^{\prime }$ and $p_1,p_2\in \mathbb{E}$, then $d_2(h_2\left(p_1\right),h_2\left(p_2\right))<\lambda$. Write ${\mathbb{U}}_j=h_2^{-1}\left({\mathbb{V}}_j\right)$ for each $j\in \{1,2\}$. By hypothesis, $h_2$ is an onto map. Consequently, ${\mathbb{U}}_j(j\in \{1,2\})$ are nonempty open sets. Then, for each $j\in \{1,2\}$, there is a $p_j\in {\mathbb{U}}_j$ such that

$$d_1({(h_1\circ h_2)}^m\left(p_1\right),{(h_1\circ h_2)}^m\left(p_2\right))<{\lambda }^{\prime }$$

for some $m\in \{1,2,\cdots \}$. Then,

$$d_2(h_2\circ {(h_1\circ h_2)}^m\left(p_1\right),h_2\circ {(h_1\circ h_2)}^m\left(p_2\right))<\lambda .$$

That is,

$$d_2({(h_2\circ h_1)}^m\left(r_1\right),{(h_2\circ h_1)}^m\left(r_2\right))<\lambda ,$$

where $r_j=h_2\left(p_j\right)\in {\mathbb{V}}_j$ for each $j\in \{1,2\}$. By the definition, $h_2\circ h_1$ is accessible. □

Similarly, one can easily verify that if $h_2\circ h_1$ is accessible, then $h_1\circ h_2$ is accessible.

Example 2.

Define $\mathbb{X}$, $\mathbb{Y}$, $h_1$, and $h_2$ as being the same as in Example 1. Then, $h_1\circ h_2$ and $h_2\circ h_1$ are accessible.

Proof. 

As one knows, tent map $\lambda \left(p\right)=1-|1-2x|$ for any $x\in [0,1]$ is topologically mixing. By [33], ${\lambda }^2$ is accessible. Clearly, in this example, one has $h_1\circ h_2={\lambda }^2$. This implies that $h_1\circ h_2$ is accessible. Then, by the proof of Lemma 2.2 in [18] and the definition of accessibility, one can easily verify that $h_2\circ h_1$ is also accessible. □

Lemma 4.

$h_1\circ h_2$ is collectively accessible if and only if $h_2\circ h_1$ is as well.

Proof. 

Assume that $h_1\circ h_2$ is collectively accessible, and that ${\mathbb{V}}_i^{\left(1\right)}\subset \mathbb{F}$ and ${\mathbb{V}}_j^{\left(1\right)}\subset \mathbb{F}$ are nonempty and open for each $i\in \{1,2,\cdots ,s\}$ and each $j\in \{1,2,\cdots ,t\}$. Write ${\mathbb{U}}_i^{\left(1\right)}=h_2^{-1}\left({\mathbb{V}}_i^{\left(1\right)}\right)$ and ${\mathbb{U}}_j^{\left(2\right)}=h_2^{-1}\left({\mathbb{V}}_j^{\left(2\right)}\right)$ for each $i\in \{1,2,\cdots ,s\}$ and each $j\in \{1,2,\cdots ,t\}$. By hypothesis, $h_2$ is an onto map. So, ${\mathbb{U}}_i^{\left(1\right)}$ ($i\in \{1,2,\cdots ,s\}$) and ${\mathbb{U}}_j^{\left(2\right)}$ ($j\in \{1,2,\cdots ,t\}$) are nonempty open sets. Set $\lambda >0$. Because $h_2$ is uniformly continuous, then there is a ${\lambda }^{\prime }>0$ such that if $d_1(a,b)<{\lambda }^{\prime }$ for $a,b\in \mathbb{E}$, then $d_2(h_2\left(a\right),h_2\left(b\right))<\lambda$. Then, by hypothesis, there are $p_i^{\left(1\right)}\in {\mathbb{U}}_i^{\left(1\right)}$ for each $i\in \{1,2,\cdots ,s\}$, $p_j^{\left(2\right)}\in {\mathbb{U}}_j^{\left(2\right)}$ and each $j\in \{1,2,\cdots ,t\}$, and some integer $m>0$, such that one of the following is true.

(1) There is an $i_0\in \{1,2,\cdots ,s\}$, such that

$$d_1({(h_1\circ h_2)}^m\left(p_{i_0}^{\left(1\right)}\right),{(h_1\circ h_2)}^m\left(p_j^{\left(2\right)}\right))<{\lambda }^{\prime }$$

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

(2) There is a $j_0\in \{1,2,\cdots ,t\}$ such that

$$d_1({(h_1\circ h_2)}^m\left(p_i^{\left(1\right)}\right),{(h_1\circ h_2)}^m\left(p_{j_0}^{\left(2\right)}\right))<{\lambda }^{\prime }$$

for every $i\in \{1,2,\cdots ,s\}$. □

Then one has

$$d_2(h_2\circ {(h_1\circ h_2)}^m\left(p_{i_0}^{\left(1\right)}\right),h_2\circ {(h_1\circ h_2)}^m\left(p_j^{\left(2\right)}\right))<\lambda (j\in \{1,2,\cdots ,t\}),$$

or

$$d_2(h_2\circ {(h_1\circ h_2)}^m\left(p_i^{\left(1\right)}\right),h_2\circ {(h_1\circ h_2)}^m\left(p_{j_0}^{\left(2\right)}\right))<\lambda (i\in \{1,2,\cdots ,s\}).$$

That is,

$$d_2({(h_2\circ h_1)}^m\left(r_{i_0}^{\left(1\right)}\right),{(h_2\circ h_1)}^m\left(r_j^{\left(2\right)}\right))<\lambda (j\in \{1,2,\cdots ,t\})$$

or

$$d_2({(h_2\circ h_1)}^m\left(r_i^{\left(1\right)}\right),{(h_2\circ h_1)}^m\left(r_{j_0}^{\left(2\right)}\right))<\lambda (i\in \{1,2,\cdots ,s\}),$$

where $r_i^{\left(1\right)}=h_2\left(p_i^1\right)\in {\mathbb{V}}_j^{\left(1\right)}$ for each $i\in \{1,2,\cdots ,s\}$ and $r_j^{\left(2\right)}=h_2\left(p_j^2\right)\in {\mathbb{V}}_j^{\left(2\right)}$ for every $j\in \{1,2,\cdots ,t\}$. By the definition, $h_2\circ h_1$ is collectively accessible.

Similarly, by hypothesis one can easily verify that if $h_2\circ h_1$ is collectively accessible, then $h_1\circ h_2$ is collectively accessible.

Let $H(p,r)=(h_1\left(r\right),h_2\left(p\right))(p\in \mathbb{E},r\in \mathbb{F})$ be a Cournot map on $\mathbb{E}\times \mathbb{F}$. In [17], the authors proved that H is Li–Yorke sensitive if and only if one of the maps $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ is as well. Inspired by this result, the following conclusions are obtained.

Theorem 1.

If H is onto, then H is sensitive if and only if $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ are sensitive.

Proof. 

By Theorem 2.2 in [18], H is sensitive if and only if either $H^2{|}_{{\mathbb{M}}_1}$ or $H^2{|}_{{\mathbb{M}}_2}$ is sensitive. By Lemma 2.1 in [18], H is sensitive if and only if $H^2=(h_1\circ h_2)\times (h_2\circ h_1)$ is sensitive. By Lemma 1 in [31], $H^2$ is sensitive if and only if either $h_1\circ h_2$ or $h_2\circ h_1$ is sensitive. Consequently, H is sensitive if and only if $h_1\circ h_2$ and $h_2\circ h_1$ are sensitive. By the proof of Theorem 2.2 in [18], Theorem 1 is true. □

Remark 2.

For Theorem 1, it is necessary to assume that $h_1:\mathbb{F}\to \mathbb{E}$ and $h_2:\mathbb{E}\to \mathbb{F}$ are continuous onto maps.

Example 3.

Define $\mathbb{X}$, $\mathbb{Y}$, $h_1$, and $h_2$ as being the same as Example 1. Then the following conclusions hold:

(1) $H^2{|}_{{\mathbb{M}}_2}$ is sensitive.

(2) $H^2{|}_{{\mathbb{M}}_1}$ is not sensitive.

Proof. 

By hypothesis, the proof of Theorem 2.2 in [18] and Example 1, one can easily deduce that $H^2{|}_{{\mathbb{M}}_2}$ is sensitive. By hypothesis, Example 1, the uniform continuity of $h_1$, and the definitions of sensitivity and ${\mathbb{M}}_1$, one can easily prove that $H^2{|}_{{\mathbb{M}}_1}$ is not sensitive. □

Lemma 5.

The Cournot map H is collectively sensitive if and only if $H^2$ is as well.

Proof. 

Clearly, if $H^2$ is collectively sensitive, then so is H. If H is collectively sensitive with the collective sensitivity constant ${\lambda }^{\prime }>0$. By the uniform continuity of H, there is a $\lambda \in (0,{\lambda }^{\prime })$ such that, $d(p_1,p_2)\le \lambda$ for $p_1,p_2\in \mathbb{E}$ implies $d(H\left(p_1\right),H\left(p_2\right))\le {\lambda }^{\prime }$. By the definition, for any $\epsilon >0$ and any m distinct points $p_1^{\left(1\right)},p_2^{\left(1\right)},\cdots ,p_m^{\left(1\right)}\in \mathbb{E}$, there are m distinct points $p_1^{\left(2\right)},p_2^{\left(2\right)},\cdots ,p_m^{\left(2\right)}\in \mathbb{E}$, such that $d(p_i^{\left(1\right)},p_i^{\left(2\right)})<\epsilon$ for all $1\le j\le m$. Furthermore, for some integer $n>0$ and any $i\in \{1,2,\cdots ,m\}$, one has $d(H^n\left(p_{i_0}^{\left(1\right)}\right),H^n\left(p_i^{\left(2\right)}\right))>{\lambda }^{\prime }$ for some $i_0\in \{1,2,\cdots ,m\}$ or $d(H^n\left(p_i^{\left(1\right)}\right),H^n\left(p_{i_0}^{\left(2\right)}\right))>{\lambda }^{\prime }$ for some $i_0\in \{1,2,\cdots ,m\}$ and any $i\in \{1,2,\cdots ,m\}$. Then, it is easily seen that if $n=2m+1$ then $d(H^{2m}\left(p_{i_0}^{\left(1\right)}\right),H^{2m}\left(p_i^{\left(2\right)}\right))>{\lambda }^{\prime }>\lambda$ or $d(H^{2m}\left(p_i^{\left(1\right)}\right),H^{2m}\left(p_{i_0}^{\left(2\right)}\right))>{\lambda }^{\prime }>\lambda$ for some $i_0\in \{1,2,\cdots ,m\}$ and any $i\in \{1,2,\cdots ,m\}$, and that if $n=2m$, then $d(H^{2m}\left(p_{i_0}^{\left(1\right)}\right),H^{2m}\left(p_i^{\left(2\right)}\right))>\lambda$ or $d(H^{2m}\left(p_i^{\left(1\right)}\right),$$H^{2m}\left(p_{i_0}^{\left(2\right)}\right))>\lambda$ for some $i_0\in \{1,2,\cdots ,m\}$ and any $i\in \{1,2,\cdots ,m\}$. By the definition, $H^2$ is collectively sensitive. □

Lemma 6.

Assume that the product metric ρ on the product space $\mathbb{E}\times \mathbb{F}$ are defined as $\rho ((p_1,r_1),$$(p_2,r_2))$$=max\left\{\right|p_2-p_1|,|r_2-r_1\left|\right\}$, where $|·|$ denotes 1-norm on $\mathbb{R}$, i.e., absolute value. The product map $f\times g$ of $f:\mathbb{E}\to \mathbb{E}$ and $g:\mathbb{F}\to \mathbb{F}$ is defined as $(f\times g)(p,r)=\left(f\right(p),g(r\left)\right)$ for any $p\in \mathbb{E}$ and any $r\in \mathbb{F}$.

If $f:\mathbb{E}\to \mathbb{E}$ and $g:\mathbb{F}\to \mathbb{F}$ are not necessarily continuous maps, then the following holds.

(1) If f or g is collectively sensitive, then $f\times g:\mathbb{E}\times \mathbb{F}\to \mathbb{E}\times \mathbb{F}$ is collectively sensitive.

(2) If $f\times g:\mathbb{E}\times \mathbb{F}\to \mathbb{E}\times \mathbb{F}$ is collectively sensitive, then at least one of f and g is collectively sensitive.

Proof. 

By the definition, the method of the proofs of Lemma 1 in [32] and Lemma 5, one easily verifies that Lemma 6 is true. □

Lemma 7.

A continuous map $h:E\to E$ is collectively accessible if and only if $h^2$ is as well.

Proof. 

By the definition, the method of the proof of Lemma 2.2 in [18] and Lemma 5, one can easily verify that Lemma 7 is true. □

Theorem 2.

H is collectively sensitive if and only if at least one of the maps $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ is collectively sensitive.

Proof. 

If H is collectively sensitive, then by Lemma 5, $H^2$ is collectively sensitive. Let the collective sensitivity constant of $H^2$ is $\lambda >0$. For any m distinct points ${\xi }_1=(p_1,h_2\left(p_1\right)),$${\xi }_2=(p_2,h_2\left(p_2\right)),\cdots ,{\xi }_m=(p_m,h_2\left(p_m\right))\in {\mathbb{M}}_2$ and any $\epsilon >0$, there are m distinct points ${\gamma }_j=(p_j,r_j)\in \mathbb{E}\times \mathbb{F}$ with $\rho ({\xi }_j,{\gamma }_j)<\epsilon$ ($j\in \{1,2,\cdots ,n\}$) and some $n\in \mathbb{N}$ such that $\rho (H^{2n}\left({\xi }_{j_0}\right),H^{2n}\left({\gamma }_j\right))>\lambda$ for some integer $j_0\in \{1,2,\cdots ,m\}$ and any $j\in \{1,2,\cdots ,m\}$, or $\rho (H^{2n}\left({\xi }_j\right),H^{2n}\left({\gamma }_{j_0}\right))>\lambda$ for some integer $j_0\in \{1,2,\cdots ,m\}$ and any $j\in \{1,2,\cdots ,m\}$. Because $H^2=(h_1\circ h_2)\times (h_2\circ h_1)$, by Lemma 6, at least one of the maps $h_1\circ h_2$ and $h_2\circ h_1$ must be collectively sensitive. Without loss of generality, one may assume that $h_1\circ h_2$ is collectively sensitive with the collective sensitivity constant $0<{\lambda }^{\prime }<\lambda$. For any $p_1^{\left(1\right)},p_2^{\left(1\right)},\cdots ,p_m^{\left(1\right)}\in \mathbb{E}$ and any $\epsilon >0$, there are points $p_1^{\left(2\right)},p_2^{\left(2\right)},\cdots ,p_m^{\left(2\right)}\in \mathbb{E}$ with $|p_i^{\left(1\right)}-p_i^{\left(2\right)}|<\epsilon$ for any $i\in \{1,2,\cdots ,m\}$. In addition, for some integer $n>0$, $|{(h_1\circ h_2)}^n\left(p_{i_0}^{\left(1\right)}\right)-{(h_1\circ h_2)}^n\left(p_i^{\left(2\right)}\right)|>{\lambda }^{\prime }$ for some $i_0\in \{1,2,\cdots ,m\}$ and any $i\in \{1,2,\cdots ,m\}$, or $|{(h_1\circ h_2)}^n\left(p_i^{\left(1\right)}\right)-{(h_1\circ h_2)}^n\left(p_{i_0}^{\left(2\right)}\right)|>{\lambda }^{\prime }$ for some $i_0\in \{1,2,\cdots ,m\}$ and any $i\in \{1,2,\cdots ,m\}$. Because $h_2$ is uniformly continuous, there is a ${\epsilon }_1\in (0,\epsilon )$ such that $|h_2\left(a_1\right)-h_2\left(a_2\right)|<\epsilon$ for any $a_1,a_2\in \mathbb{E}$ with $|a_1-a_2|<{\epsilon }_1$. Clearly, $\rho ((p_i^{\left(1\right)},h_2\left(p_i^{\left(1\right)}\right)),(p_i^{\left(2\right)},h_2\left(p_i^{\left(2\right)}\right)))<\epsilon$ for any $i\in \{1,2,\cdots ,m\}$. Consequently,

$$\rho (H^{2n}(p_{i_0}^{\left(1\right)},h_2\left(p_{i_0}^{\left(1\right)}\right)),H^{2n}(p_i^{\left(2\right)},h_2\left(p_i^{\left(2\right)}\right)))\ge |{(h_1\circ h_2)}^n\left(p_{i_0}^{\left(1\right)}\right)-{(h_1\circ h_2)}^n\left(p_i^{\left(2\right)}\right)|>{\lambda }^{\prime }$$

for some $i_0\in \{1,2,\cdots ,m\}$ and any $i\in \{1,2,\cdots ,m\}$, or

$$\rho (H^{2n}(p_i^{\left(1\right)},h_2\left(p_i^{\left(1\right)}\right)),H^{2n}(p_{i_0}^{\left(2\right)},h_2\left(p_{i_0}^{\left(2\right)}\right)))\ge |{(h_1\circ h_2)}^n\left(p_i^{\left(1\right)}\right)-{(h_1\circ h_2)}^n\left(p_{i_0}^{\left(2\right)}\right)|>{\lambda }^{\prime }$$

for some $i_0\in \{1,2,\cdots ,m\}$ and any $i\in \{1,2,\cdots ,m\}$. By the definition, $H^2{|}_{{\mathbb{M}}_2}$ is collectively sensitive.

Conversely, assume that $H^2{|}_{{\mathbb{M}}_2}$ is collectively sensitive with a collective sensitivity constant $\lambda >0$. By the uniform continuity of $h_2$, there is a ${\lambda }^{\prime }:0<{\lambda }^{\prime }<\lambda$ such that, for any two points $a_1,a_2\in \mathbb{E}$, $|a_1-a_2|\le {\lambda }^{\prime }$ means $|h_2\left(a_1\right)-h_2\left(a_2\right)|\le \lambda$. Consequently, for any $i\in \{1,2,\cdots ,m\}$ and any $p_i^{\left(1\right)},p_i^{\left(2\right)}\in \mathbb{E}$ with $|p_i^{\left(1\right)}-p_i^{\left(2\right)}|\le {\lambda }^{\prime }$, if

$$\rho (H^{2n}(p_{i_0}^{\left(1\right)},h_2\left(p_{i_0}^{\left(1\right)}\right)),H^{2n}(p_i^{\left(2\right)},h_2\left(p_i^{\left(2\right)}\right)))>\lambda$$

for some $n>0$, some $i_0\in \{1,2,\cdots ,m\}$ and any $i\in \{1,2,\cdots ,m\}$, or

$$\rho (H^{2n}(p_i^{\left(1\right)},h_2\left(p_i^{\left(1\right)}\right)),H^{2n}(p_{i_0}^{\left(2\right)},h_2\left(p_{i_0}^{\left(2\right)}\right)))>\lambda$$

for some $n>0$, some $i_0\in \{1,2,\cdots ,m\}$ and any $i\in \{1,2,\cdots ,m\}$, then

$$|{(h_1\circ h_2)}^n\left(p_{i_0}^{\left(1\right)}\right)-{(h_1\circ h_2)}^n\left(p_i^{\left(2\right)}\right)\left)\right|>{\lambda }^{\prime }$$

for some $n>0$, some $i_0\in \{1,2,\cdots ,m\}$, and any $i\in \{1,2,\cdots ,m\}$, or

$$|{(h_1\circ h_2)}^n\left(p_i^{\left(1\right)}\right)-{(h_1\circ h_2)}^n\left(p_{i_0}^{\left(2\right)}\right)\left)\right|>{\lambda }^{\prime }$$

for some $n>0$, some $i_0\in \{1,2,\cdots ,m\}$, and any $i\in \{1,2,\cdots ,m\}$. Thus, $h_1\circ h_2$ is collectively sensitive. By Theorem 3.1 in [17] and Lemma 6, H is collectively sensitive. □

Theorem 3.

If H is onto, then H is collectively sensitive if and only if $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ are collectively sensitive.

Proof. 

By the proof of Theorem 2.2 in [18], one can easily prove that H is collectively sensitive if and only if either $H^2{|}_{{\mathbb{M}}_1}$ or $H^2{|}_{{\mathbb{M}}_2}$ is collectively sensitive. By the proof of Lemma 2.1 in [18], one has that H is collectively sensitive if and only if $H^2$ is as well. Because $H^2=(h_1\circ h_2)\times (h_2\circ h_1)$, by the proof of Lemma 1 in [31], one can easily prove that $H^2$ is collectively sensitive if and only if either $h_1\circ h_2$ or $h_2\circ h_1$ is collectively sensitive. By Lemma 2, $H^2$ is collectively sensitive if and only if $h_1\circ h_2$ and $h_2\circ h_1$ are collectively sensitive. By the proof of Theorem 2.2 in [18], Theorem 3 is true. □

Theorem 4.

(1) If H is collectively accessible, then both $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ are collectively accessible.

(2) Assume that $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ are collectively accessible. For any $\epsilon >0$ and any given integers $s,t>0$, if $\mid {(h_2\circ h_1)}^n\left(p_j^{\left(1\right)}\right)-{(h_2\circ h_1)}^n\left(p_i^{\left(2\right)}\right)\mid <\epsilon$ (i.e., the absolute value of the real number ${(h_2\circ h_1)}^n\left(p_j^{\left(1\right)}\right)-{(h_2\circ h_1)}^n\left(p_i^{\left(2\right)}\right)$) and $\mid {(h_1\circ h_2)}^m\left(r_j^{\left(1\right)}\right)-{(h_1\circ h_2)}^m\left(r_i^{\left(2\right)}\right)\mid <\epsilon$ for some integers $n,m>0$, any $j\in \{1,2,\cdots ,t\}$, and any $i\in \{1,2,\cdots ,s\}$ implies that there is an integer $l(n,m,\epsilon )>0$ satisfies

$$\mid {(h_2\circ h_1)}^{l(n,m,\epsilon )}\left(p_j^{\left(1\right)}\right)-{(h_2\circ h_1)}^{l(n,m,\epsilon )}\left(p_i^{\left(2\right)}\right)\mid <\epsilon$$

and

$$\mid {(h_1\circ h_2)}^{l(n,m,\epsilon )}\left(r_j^{\left(1\right)}\right)-{(h_1\circ h_2)}^{l(n,m,\epsilon )}\left(r_i^{\left(2\right)}\right)\mid <\epsilon$$

for any $j\in \{1,2,\cdots ,t\}$ and any $i\in \{1,2,\cdots ,s\}$, then H is collectively accessible.

Proof 

(1) Assume that H is collectively accessible. $U_1^{\left(1\right)},U_2^1,\cdots ,U_s^{\left(1\right)}$, $U_1^{\left(2\right)},U_2^{\left(2\right)},\cdots ,U_t^{\left(2\right)}$$\subset \mathbb{E}$, and $V_1^{\left(1\right)},V_2^{\left(1\right)},\cdots ,V_s^{\left(1\right)},V_1^{\left(2\right)},V_2^{\left(2\right)},\cdots ,V_t^{\left(2\right)}\subset \mathbb{F}$ are nonempty open sets.

By the definition, one can easily prove that, for any continuous maps $f:\mathbb{E}\to \mathbb{E}$ and $g:\mathbb{F}\to \mathbb{F}$, if product map $f\times g$ is collectively accessible then both f and g are collectively accessible. Consequently, for any $\epsilon >0$, $\mid h_2^n\left(p_j^{\left(1\right)}\right)-h_2^n\left(p_i^{\left(2\right)}\right)\mid <\epsilon$ and $\mid h_1^m\left(r_j^{\left(1\right)}\right)-h_1^m\left(r_i^{\left(2\right)}\right)\mid <\epsilon$ for some integers $n,m>0$, any $j\in \{1,2,\cdots ,t\}$ and any $i\in \{1,2,\cdots ,s\}$ implies that there is an integer $l(n,m,\epsilon )>0$ that satisfies

$$\mid h_2^{l(n,m,\epsilon )}\left(p_j^{\left(1\right)}\right)-h_2^{l(n,m,\epsilon )}\left(p_i^{\left(2\right)}\right)\mid <\epsilon$$

and

$$\mid h_1^{l(n,m,\epsilon )}\left(r_j^{\left(1\right)}\right)-h_1^{l(n,m,\epsilon )}\left(r_j^{\left(2\right)}\right)\mid <\epsilon$$

for any $j\in \{1,2,\cdots ,t\}$ and any $i\in \{1,2,\cdots ,s\}$. Then, $h_1\times h_2$ is collectively accessible. By the definition, for any $\epsilon >0$ and any nonempty subsets $U_1^{\left(1\right)}\times V_1^{\left(1\right)},\cdots ,U_s^{\left(1\right)}\times V_s^{\left(1\right)},U_1^{\left(2\right)}\times V_1^{\left(2\right)},\cdots ,U_t^{\left(2\right)}\times V_t^{\left(2\right)}\subset \mathbb{E}\times \mathbb{F}$, there are $(p_1^{\left(1\right)},r_1^{\left(1\right)})\in U_1^{\left(1\right)}\times V_1^{\left(1\right)},\cdots ,(p_s^{\left(1\right)},r_s^{\left(1\right)})\in U_s^{\left(1\right)}\times V_s^{\left(1\right)},(p_1^{\left(2\right)},r_1^{\left(2\right)})\in U_1^{\left(2\right)}\times V_1^{\left(2\right)},\cdots ,(p_t^{\left(2\right)},r_t^{\left(2\right)})\in U_t^{\left(2\right)}\times V_t^{\left(2\right)}$, and some integers $m,n>0$ satisfies

$$\rho ({(h_1\times h_2)}^m(p_{j_0}^{\left(1\right)},r_{j_0}^{\left(1\right)}),{(h_1\times h_2)}^m(p_i^{\left(2\right)},r_i^{\left(2\right)}))<\epsilon$$

for some $j_0\in \{1,\cdots ,s\}$ and any $i\in \{1,\cdots ,t\}$ or

$$\rho ({(h_1\times h_2)}^m(p_j^{\left(1\right)},r_j^{\left(1\right)}),{(h_1\times h_2)}^m(p_{i_0}^{\left(2\right)},r_{i_0}^{\left(2\right)}))<\epsilon$$

for some $i_0\in \{1,\cdots ,t\}$ and any $j\in \{1,\cdots ,s\}$. Consequently, one has that $\mid h_1^m\left(p_{j_0}^{\left(1\right)}\right)-h_1^m\left(p_i^{\left(2\right)}\right)\mid <\epsilon$ and $\mid h_2^n\left(r_{j_0}^{\left(1\right)}\right)-h_2^n\left(r_i^{\left(2\right)}\right)\mid <\epsilon$ for any $i\in \{1,\cdots ,t\}$, or $\mid h_1^m\left(p_j^{\left(1\right)}\right)-h_1^m\left(p_{i_0}^{\left(2\right)}\right)\mid <\epsilon$ and $\mid h_2^n\left(r_j^{\left(1\right)}\right)-h_2^n\left(r_{i_0}^{\left(2\right)}\right)\mid <\epsilon$ for any $j\in \{1,\cdots ,s\}$. By the definition, $h_1$ and $h_2$ are accessible. If $h_1$ and $h_2$ are collectively accessible, then, for any nonempty open subsets $U_1^{\left(1\right)},\cdots ,U_s^{\left(1\right)},U_1^{\left(2\right)},\cdots ,U_t^{\left(2\right)}\subset \mathbb{E}$, any nonempty open subsets $V_1^{\left(1\right)},\cdots ,V_s^{\left(1\right)},V_1^{\left(2\right)},\cdots ,$$V_t^{\left(2\right)}\subset \mathbb{F}$ and any $\epsilon >0$, there are $p_j^{\left(1\right)}\in U_j^{\left(1\right)}$ and $r_j^{\left(1\right)}\in V_j^{\left(1\right)}$ for any $j\in \{1,\cdots ,s\}$, $p_i^{\left(2\right)}\in U_i^{\left(2\right)}$ and $r_i^{\left(2\right)}\in V_i^{\left(2\right)}$ for any $i\in \{1,\cdots ,t\}$, and two integers $m,n>0$, such that

$$\mid h_1^m\left(p_{j_1}^{\left(1\right)}\right)-h_1^m\left(p_i^{\left(2\right)}\right)\mid <\epsilon \mathrm{and}\mid h_2^n\left(r_{j_2}^{\left(1\right)}\right)-h_2^n\left(r_i^{\left(2\right)}\right)\mid <\epsilon$$

for any $i\in \{1,\cdots ,t\}$ and some $j_1,j_2\in \{1,\cdots ,s\}$, or

$$\mid h_1^m\left(p_j^{\left(1\right)}\right)-h_1^m\left(p_{i_1}^{\left(2\right)}\right)\mid <\epsilon \mathrm{and}\mid h_2^n\left(r_j^{\left(1\right)}\right)-h_2^n\left(r_{i_2}^{\left(2\right)}\right)\mid <\epsilon$$

for some $i_1,i_2\in \{1,\cdots ,t\}$ and any $j\in \{1,\cdots ,s\}$. By hypothesis,

$$\mid h_1^{l(n,m,\epsilon )}\left(p_{j_1}^{\left(1\right)}\right)-h_1^{l(n,m,\epsilon )}\left(p_i^{\left(2\right)}\right)\mid <\epsilon \mathrm{and}\mid h_2^{l(n,m,\epsilon )}\left(r_{j_2}^{\left(1\right)}\right)-h_2^{l(n,m,\epsilon )}\left(r_i^{\left(2\right)}\right)\mid <\epsilon$$

for any $i\in \{1,\cdots ,t\}$, or

$$\mid h_1^{l(n,m,\epsilon )}\left(p_j^{\left(1\right)}\right)-h_1^{l(n,m,\epsilon )}\left(p_{i_1}^{\left(2\right)}\right)\mid <\epsilon \mathrm{and}\mid h_2^{l(n,m,\epsilon )}\left(r_j^{\left(1\right)}\right)-h_2^{l(n,m,\epsilon )}\left(r_{i_2}^{\left(2\right)}\right)\mid <\epsilon$$

for any $j\in \{1,\cdots ,s\}$. Consequently,

$$\rho ({(h_1\times h_2)}^{l(n,m,\epsilon )}(p_{j_1}^{\left(1\right)},r_{j_2}^{\left(1\right)}),{(h_1\times h_2)}^{l(n,m,\epsilon )}(p_i^{\left(2\right)},r_i^{\left(2\right)}))<\epsilon$$

for any $i\in \{1,\cdots ,t\}$ or

$$\rho ({(h_1\times h_2)}^{l(n,m,\epsilon )}(p_j^{\left(1\right)},r_j^{\left(1\right)}),{(h_1\times h_2)}^{l(n,m,\epsilon )}(p_{i_1}^{\left(2\right)},r_{i_2}^{\left(2\right)}))<\epsilon$$

for any $j\in \{1,\cdots ,s\}$. By the definition, $h_1\times h_2$ is collectively accessible.

By Lemma 7, $H^2$ is collectively accessible. By the above argument, $h_1\circ h_2$ and $h_2\circ h_1$ are collectively accessible. Because $h_1\circ h_2$ is collectively accessible, for any nonempty open subsets $U_1^{\left(1\right)},\cdots ,U_s^{\left(1\right)},U_1^{\left(2\right)},\cdots ,U_t^{\left(2\right)}\subset \mathbb{E}$ and $V_1^{\left(1\right)},\cdots ,V_s^{\left(1\right)},V_1^{\left(2\right)},\cdots ,V_t^{\left(2\right)}\subset \mathbb{F}$ satisfies

$$\left(U_j^{\left(1\right)}\times V_j^{\left(1\right)}\right)\cap {\mathbb{M}}_2\ne \varnothing (j\in \{1,\cdots ,s\})$$

and

$$\left(U_i^{\left(2\right)}\times V_i^{\left(2\right)}\right)\cap {\mathbb{M}}_2\ne \varnothing (i\in \{1,\cdots ,t\}),$$

one has that $U_j^{\left(1\right)}\cap h_2^{-1}\left(V_j^{\left(1\right)}\right)\ne \varnothing$ for any $j\in \{1,\cdots ,s\}$, and that $U_i^{\left(2\right)}\cap h_2^{-1}\left(V_i^{\left(2\right)}\right)\ne \varnothing$ for any $i\in \{1,\cdots ,t\}$. By the definition, for any $\lambda >0$, any $j\in \{1,\cdots ,s\}$ and $i\in \{1,\cdots ,t\}$, there are $p_j^{\left(1\right)}\left(\lambda \right)\in U_j^{\left(1\right)}\cap h_2^{-1}\left(V_j^{\left(1\right)}\right)$, $p_i^{\left(2\right)}\left(\lambda \right)\in U_i^{\left(2\right)}\cap h_2^{-1}\left(V_i^{\left(2\right)}\right)$, and an integer $m\left(\lambda \right)>0$ such that

$$|{(h_1\circ h_2)}^{m\left(\lambda \right)}\left(p_{j_0}^{\left(1\right)}\left(\lambda \right)\right)-{(h_1\circ h_2)}^{m\left(\lambda \right)}\left(p_i^{\left(2\right)}\left(\lambda \right)\right)|<\lambda$$

for any $i\in \{1,\cdots ,t\}$ and some $j_0\in \{1,\cdots ,s\}$, or

$$|{(h_1\circ h_2)}^{m\left(\lambda \right)}\left(p_j^{\left(1\right)}\left(\lambda \right)\right)-{(h_1\circ h_2)}^{m\left(\lambda \right)}\left(p_{i_0}^{\left(2\right)}\left(\lambda \right)\right)|<\lambda$$

for some $i_0\in \{1,\cdots ,t\}$ and any $j\in \{1,\cdots ,s\}$. By the uniform continuity of $h_2$, one knows that for any $\lambda >0$, there is some ${\lambda }^{\prime }>0$ with ${\lambda }^{\prime }<\lambda$ such that $|u_1-u_2|<{\lambda }^{\prime }$ implies $|h_2\left(u_1\right)-h_2\left(u_2\right)|<\lambda$. Let $z_j^{\left(1\right)}\left({\lambda }^{\prime }\right)=(p_j^{\left(1\right)}\left({\lambda }^{\prime }\right),h_2\left(p_j^{\left(1\right)}\left({\lambda }^{\prime }\right)\right))$ for any $j\in \{1,\cdots ,s\}$ and $z_i^{\left(2\right)}\left({\lambda }^{\prime }\right)=(p_i^{\left(2\right)}\left({\lambda }^{\prime }\right),h_2\left(p_i^{\left(2\right)}\left({\lambda }^{\prime }\right)\right))$ for any $i\in \{1,\cdots ,t\}$. Then

$$z_j^{\left(1\right)}\left({\lambda }^{\prime }\right)\in \left(U_j^{\left(1\right)}\times V_j^{\left(1\right)}\right)\cap {\mathbb{M}}_2(j\in \{1,\cdots ,s\})$$

and

$$z_i^{\left(2\right)}\left({\lambda }^{\prime }\right)\in \left(U_i^{\left(2\right)}\times V_i^{\left(2\right)}\right)\cap {\mathbb{M}}_2(i\in \{1,\cdots ,t\}).$$

It is clear that

$$|{(h_1\circ h_2)}^{m\left({\lambda }^{\prime }\right)}\left(p_{j_0}^{\left(1\right)}\left({\lambda }^{\prime }\right)\right)-{(h_1\circ h_2)}^{m\left({\lambda }^{\prime }\right)}\left(p_i^{\left(2\right)}\left({\lambda }^{\prime }\right)\right)|<{\lambda }^{\prime }<\lambda (i\in \{1,\cdots ,t\}),$$

or

$$|{(h_2\circ h_1)}^{m\left({\lambda }^{\prime }\right)}\left(h_2\left(p_j^{\left(1\right)}\left({\lambda }^{\prime }\right)\right)\right)-{(h_2\circ h_1)}^{m\left({\lambda }^{\prime }\right)}\left(h_2\left(p_{i_0}^{\left(2\right)}\left({\lambda }^{\prime }\right)\right)\right)|<\lambda (j\in \{1,\cdots ,s\}).$$

Consequently,

$$\rho (H^{2m\left({\lambda }^{\prime }\right)}\left(z_{j_0}^{\left(1\right)}\left({\lambda }^{\prime }\right)\right),H^{2m\left({\lambda }^{\prime }\right)}\left(z_i^{\left(2\right)}\left({\lambda }^{\prime }\right)\right))<\lambda (i\in \{1,\cdots ,t\})$$

or

$$\rho (H^{2m\left({\lambda }^{\prime }\right)}\left(z_j^{\left(1\right)}\left({\lambda }^{\prime }\right)\right),H^{2m\left({\lambda }^{\prime }\right)}\left(z_{i_0}^{\left(2\right)}\left({\lambda }^{\prime }\right)\right))<\lambda (j\in \{1,\cdots ,s\}).$$

This implies that $H^2{|}_{{\mathbb{M}}_2}$ is collectively accessible. Similarly, one can prove that if $h_2\circ h_1$ is collectively accessible then so is $H^2{|}_{{\mathbb{M}}_1}$.

(2) Now, assume that $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ are collectively accessible, and that the maps $h_1$ and $h_2$ satisfy the conditions in the assumption.

Let $H^2{|}_{{\mathbb{M}}_2}$ be collectively accessible. Obviously, for any nonempty open subsets $U_1^{\left(1\right)},\cdots$, $U_1^{\left(s\right)},U_1^{\left(2\right)},\cdots ,U_t^{\left(2\right)}\subset \mathbb{E}$, $(U_j^{\left(1\right)}\times \mathbb{F})\cap {\mathbb{M}}_2$ and $(U_i^{\left(2\right)}\times \mathbb{F})\cap {\mathbb{M}}_2$ are nonempty and open in ${\mathbb{M}}_2$ for any $i\in \{1,2,\cdots ,t\}$ and any $j\in \{1,2,\cdots ,s\}$. By the definition, for any $\lambda >0$, there are

$$p_1^{\left(1\right)}\in U_1^{\left(1\right)},\cdots ,p_s^{\left(1\right)}\in U_s^{\left(1\right)},p_1^{\left(2\right)}\in U_1^{\left(2\right)},\cdots ,p_s^{\left(2\right)}\in U_t^{\left(2\right)},$$

and an integer $m\left(\lambda \right)>0$ such that

$$\rho (H^{2m\left(\lambda \right)}(p_{j_0}^{\left(1\right)}\left(\lambda \right),h_2\left(p_{j_0}^{\left(1\right)}\left(\lambda \right)\right)),H^{2m\left(\lambda \right)}(p_i^{\left(2\right)}\left(\lambda \right),h_2\left(p_i^{\left(2\right)}\left(\lambda \right)\right)))<\lambda$$

for any $i\in \{1,\cdots ,t\}$ and some $j_0\in \{1,\cdots ,s\}$, or

$$\rho (H^{2m\left(\lambda \right)}(p_{j^{\left(1\right)}}\left(\lambda \right),h_2\left(p_j^{\left(1\right)}\left(\lambda \right)\right)),H^{2m\left(\lambda \right)}(p_{i_0}^{\left(2\right)}\left(\lambda \right),h_2\left(p_{i_0}^{\left(2\right)}\left(\lambda \right)\right)))<\lambda$$

for some $i_0\in \{1,\cdots ,t\}$ and any $j\in \{1,\cdots ,s\}$. Consequently,

$$|{(h_1\circ h_2)}^{m\left(\lambda \right)}\left(p_{j_0}^{\left(1\right)}\left(\lambda \right)\right)-{(h_1\circ h_2)}^{m\left(\lambda \right)}\left(p_i^{\left(2\right)}\left(\lambda \right)\right)|<\lambda$$

for $i\in \{1,\cdots ,t\}$ or

$$|{(h_1\circ h_2)}^{m\left(\lambda \right)}\left(p_j^{\left(1\right)}\left(\lambda \right)\right)-{(h_1\circ h_2)}^{m\left(\lambda \right)}\left(p_{i_0}^{\left(2\right)}\left(\lambda \right)\right)|<\lambda$$

for any $j\in \{1,\cdots ,s\}$. This proves that $h_1\circ h_2$ is collectively accessible. Similarly, it can be proven that $h_2\circ h_1$ is collectively accessible. By the hypothesis, the definition, and the above argument, $H^2$ is collectively accessible. By Lemma 7, H is collectively accessible. □

Remark 3.

It cannot be known whether the product map $h_1\times h_2$ is collectively accessible under the condition that $h_1$ and $h_2$ are collectively accessible, respectively.

Theorem 5.

(1) If H is collectively Kato chaotic, then at least one of $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ is collectively Kato chaotic.

(2) Assume that $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ is collectively Kato chaotic. For any $\epsilon >0$ and any given integers $s,t>0$, if $\mid {(h_2\circ h_1)}^n\left(p_j^{\left(1\right)}\right)-{(h_2\circ h_1)}^n\left(p_i^{\left(2\right)}\right)\mid <\epsilon$ and $\mid {(h_1\circ h_2)}^m\left(r_j^{\left(1\right)}\right)-{(h_1\circ h_2)}^m\left(r_i^{\left(2\right)}\right)\mid <\epsilon$ for some integers $n,m>0$, any $j\in \{1,2,\cdots ,t\}$, and any $i\in \{1,2,\cdots ,s\}$ means that there is an integer $l(n,m,\epsilon )>0$ satisfies

$$\mid {(h_2\circ h_1)}^{l(n,m,\epsilon )}\left(p_j^{\left(1\right)}\right)-{(h_2\circ h_1)}^{l(n,m,\epsilon )}\left(p_i^{\left(2\right)}\right)\mid <\epsilon$$

and

$$\mid {(h_1\circ h_2)}^{l(n,m,\epsilon )}\left(r_j^{\left(1\right)}\right)-{(\alpha \circ h_2)}^{l(n,m,\epsilon )}\left(r_i^{\left(2\right)}\right)\mid <\epsilon$$

for any $j\in \{1,2,\cdots ,t\}$ and any $i\in \{1,2,\cdots ,s\}$, then H is collectively Kato chaotic.

Proof. 

According to the definition of collective Kato’s chaos, Theorem 2, and Theorem 4, Theorem 5 is true. □

Example 4.

Let $\mathbb{X}=[0,1]$ and $\mathbb{Y}=[0,1]$. $h_1:\mathbb{Y}\to \mathbb{X}$ is defined by $h_1\left(r\right)=1-|1-2r|$ for any $r\in \mathbb{X}$, and $h_2:\mathbb{X}\to \mathbb{Y}$ is $h_2\left(p\right)=1-|1-2p|$ for any $p\in \mathbb{X}$. Then, H, $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ are collectively Kato chaotic.

Proof. 

Clearly, $h_1(=h_2)$ is a tent map. Because any tent map is topologically mixing, by Corollary 4.1 in [34], $h_1$ is collectively Kato chaotic. By the definition, the hypothesis, and Lemma 6, $h_1\times h_2$ is collectively sensitive. This implies that $H^2$ is collectively sensitive. By Lemma 5, H is collectively sensitive. By Theorem 2 and its proof, $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ are collectively sensitive. By Lemma 3.5 in [34] and the proof of Theorem 3.4 in [34], one can easily prove that $h_1\times h_2$ is collectively accessible, which implies $H^2$ is collectively accessible. By Lemma 7, H is collectively accessible. By hypothesis and Lemma 3.5 in [34], one can easily prove that $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ are accessible. Thus, H, $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ are collectively Kato chaotic. □

Theorem 6.

If H is onto, then H is sensitive if and only if so is $H^2{|}_{{\mathbb{M}}_1\cup {\mathbb{M}}_2}$.

Proof. 

Assume that H is sensitive. By Lemma 2.1 in [18], $H^2$ is sensitive. Therefore, by this hypothesis, the definition of $H^2{|}_{{\mathbb{M}}_1\cup {\mathbb{M}}_2}$, and the uniform continuity of $h_1$ and $h_2$, $H^2{|}_{{\mathbb{M}}_1\cup {\mathbb{M}}_2}$ is sensitive.

Assume that $H^2{|}_{{\mathbb{M}}_1\cup {\mathbb{M}}_2}$ is sensitive. Then, by this hypothesis, the definition of $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$, the sensitivity of $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$, and Theorem 2.2 in [18], H is sensitive. □

Example 5.

Define $\mathbb{X}$, $\mathbb{Y}$, $h_1$, and $h_2$ are the same as Example 1. Then the following hold:

(1) H is sensitive.

(2) $H^2{|}_{{\mathbb{M}}_1\cup {\mathbb{M}}_2}$ is not sensitive.

Proof. 

By hypothesis, Example 3.1, Lemma 1 in [32], and Theorem 2.2 in [18], one can easily prove that H is sensitive. By the hypothesis and Example 1, one can easily prove that $H^2{|}_{{\mathbb{M}}_1}$ is not sensitive. Consequently, by the definitions of $H^2{|}_{{\mathbb{M}}_1}$, $H^2{|}_{{\mathbb{M}}_2}$ and $H^2{|}_{{\mathbb{M}}_1\cup {\mathbb{M}}_2}$, $H^2{|}_{{\mathbb{M}}_1\cup {\mathbb{M}}_2}$ is not sensitive. □

Remark 4.

Example 5 shows that it is necessary to assume that $h_1:\mathbb{F}\to \mathbb{E}$ and $h_2:\mathbb{E}\to \mathbb{F}$ are continuous onto maps for Theorem 3.6.

Theorem 7.

If H is onto, then H is collectively sensitive if and only if $H^2{|}_{{\mathbb{M}}_1\cup {\mathbb{M}}_2}$ is as well.

Proof. 

Assume that H is collectively sensitive. By Lemma 2.1 in [18], $H^2$ is collectively sensitive. Therefore, by the hypothesis, the definition of $H^2{|}_{{\mathbb{M}}_1\cup {\mathbb{M}}_2}$, and the uniform continuity of $h_1$ and $h_2$, $H^2{|}_{{\mathbb{M}}_1\cup {\mathbb{M}}_2}$ is collectively sensitive.

Assume that $H^2{|}_{{\mathbb{M}}_1\cup {\mathbb{M}}_2}$ is collectively sensitive. Then, by the hypothesis, the definition of $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$, $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ are collectively sensitive. By Theorem 2.2 in [18], H is collectively sensitive. □

Theorem 8.

If H is onto, and for any $\epsilon >0$, any given integers $s,t>0$, there exist some integers $n,m>0$, such that $\mid {(h_2\circ h_1)}^n\left(p_j^{\left(1\right)}\right)-{(h_2\circ h_1)}^n\left(p_i^{\left(2\right)}\right)\mid <\epsilon$ ($p_j^{\left(1\right)},p_i^{\left(2\right)}\in \mathbb{E}$) and $\mid {(h_1\circ h_2)}^m\left(r_j^{\left(1\right)}\right)-{(h_1\circ h_2)}^m\left(r_i^{\left(2\right)}\right)\mid <\epsilon$ ($r_j^{\left(1\right)},r_i^{\left(2\right)}\in \mathbb{F}$) for any $j\in \{1,2,\cdots ,t\}$ and $i\in \{1,2,\cdots ,s\}$, then there is an integer $l(n,m,\epsilon )>0$ satisfies

$$\mid {(h_2\circ h_1)}^{l(n,m,\epsilon )}\left(p_j^{\left(1\right)}\right)-{(h_2\circ h_1)}^{l(n,m,\epsilon )}\left(p_i^{\left(2\right)}\right)\mid <\epsilon$$

and

$$\mid {(h_1\circ h_2)}^{l(n,m,\epsilon )}\left(r_j^{\left(1\right)}\right)-{(h_1\circ h_2)}^{l(n,m,\epsilon )}\left(r_i^{\left(2\right)}\right)\mid <\epsilon$$

for any $j\in \{1,2,\cdots ,t\}$ and any $i\in \{1,2,\cdots ,s\}$, then H is collectively accessible if and only if so is $H^2{|}_{{\mathbb{M}}_1\cup {\mathbb{M}}_2}$.

Proof. 

Assume that H is collectively accessible. By Lemma 2.2 in [18], $H^2$ is collectively accessible. Therefore, by the hypothesis, the definition of $H^2{|}_{{\mathbb{M}}_1\cup {\mathbb{M}}_2}$, and the uniform continuity of $h_1$ and $h_2$, $H^2{|}_{{\mathbb{M}}_1\cup {\mathbb{M}}_2}$ is collectively accessible.

Assume that $H^2{|}_{{\mathbb{M}}_1\cup {\mathbb{M}}_2}$ is collectively accessible. Then, by the hypothesis, the definition of $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$, one has that $H^2{|}_{{\mathbb{M}}_1}$ and $H^2{|}_{{\mathbb{M}}_2}$ are collectively accessible. By Theorem 4, H is collectively accessible. □

Theorem 9.

If H is onto, and for any $\epsilon >0$, any given integers $s,t>0$, there exist some integers $n,m>0$, such that $\mid {(h_2\circ h_1)}^n\left(p_j^{\left(1\right)}\right)-{(h_2\circ h_1)}^n\left(p_i^{\left(2\right)}\right)\mid <\epsilon$ ($p_j^{\left(1\right)},p_i^{\left(2\right)}\in \mathbb{E}$) and $\mid {(h_1\circ h_2)}^m\left(r_j^{\left(1\right)}\right)-{(h_1\circ h_2)}^m\left(r_i^{\left(2\right)}\right)\mid <\epsilon$ ($r_j^{\left(1\right)},r_i^{\left(2\right)}\in \mathbb{F}$) for any $j\in \{1,2,\cdots ,t\}$ and any $i\in \{1,2,\cdots ,s\}$, then there is an integer $l(n,m,\epsilon )>0$ that satisfies

$$\mid {(h_2\circ h_1)}^{l(n,m,\epsilon )}\left(p_j^{\left(1\right)}\right)-{(h_2\circ h_1)}^{l(n,m,\epsilon )}\left(p_i^{\left(2\right)}\right)\mid <\epsilon$$

and

$$\mid {(h_1\circ h_2)}^{l(n,m,\epsilon )}\left(r_j^{\left(1\right)}\right)-{(h_1\circ h_2)}^{l(n,m,\epsilon )}\left(r_i^{\left(2\right)}\right)\mid <\epsilon$$

for any $j\in \{1,2,\cdots ,t\}$ and any $i\in \{1,2,\cdots ,s\}$, and then H is collectively Kato chaotic if and only if so is $H^2{|}_{{\mathbb{M}}_1\cup {\mathbb{M}}_2}$.

Proof. 

By Theorems 7 and 8, Theorem 9 holds. □

4. Conclusions

For an onto Cournot map $H(p,r)=(h_1\left(r\right),h_2\left(p\right))(p\in \mathbb{E},r\in \mathbb{F})$ on product space $\mathbb{E}\times \mathbb{F}$, $h_1\circ h_2$ is sensitive (resp. accessible, collectively sensitive, collectively accessible) if and only if $h_2\circ h_1$ is as well. H is $\mathcal{Q}$-chaos if and only if $H^2{|}_{{\mathbb{M}}_1}$ (i.e., the restriction of the map $H^2$ to the set ${\mathbb{M}}_1$) and $H^2{|}_{{\mathbb{M}}_2}$ (i.e., the restriction of the map $H^2$ to the set ${\mathbb{M}}_2$) are $\mathcal{Q}$-chaos. Where $\mathcal{Q}$-chaos is sensitive, collectively sensitive, collectively accessible, accessible, or collectively Kato’s chaos, the set ${\mathbb{M}}_1$ is given by $\{(h_1\left(r\right),r):r\in \mathbb{F}\}$, and the set ${\mathbb{M}}_2$ is given by $\{(p,h_2\left(p\right)):p\in \mathbb{E}\}$. Moreover, Theorems 3.7-3.9 provide the necessary and sufficient conditions of that $H^2{|}_{{\mathbb{M}}_1\cup {\mathbb{M}}_2}$ is $\mathcal{Q}$-chaos.

The conclusions in this paper implies that, in the duopoly games, one can predict the complexity of the duopoly results of both sides by analyzing the behavior of one side of the market. Failure of either side to abide by market rules will lead to failure of cooperation.