Sensitivity, Shadowing Property and P-Chaos in Duopoly Games

Abstract

In this paper, we discussed the cofinite sensitivity, shadowing property (SP), P-chaos, and chain mixing of a system induced by symmetric maps (Cournot maps) $D(a,b)=\left(t\right(b),s(a\left)\right)$ over a product space $G\times H$, where $s:G\to H$, $t:H\to G$, $a\in G$, $b\in H$, G and H are closed subintervals with $G,H\subset \mathbb{R}$. The following hold: (1) D is cofinitely sensitive and equivalent to $D^2{|}_{{\Gamma }_1}$ or $D^2{|}_{{\Gamma }_2}$ being sensitive, where ${\Gamma }_1=\{(t\left(b\right),b):b\in H\}$, ${\Gamma }_2=\{(a,s\left(a\right)):a\in G\}$. (2) D possessing an SP is equivalent to both $s\circ t$ and $t\circ s$ having an SP. (3) $t\circ s$ possesses an SP if and only if $s\circ t$ does as well. (4) D is P-chaotic and equivalent to the maps $s\circ t$ and $t\circ s$ being P-chaotic. (5) If D is chain mixing, then both $D^2{|}_{{\Gamma }_1}$ and $D^2{|}_{{\Gamma }_2}$ are chain mixing. (6) If $D^2{|}_{{\Gamma }_1}$ and $D^2{|}_{{\Gamma }_2}$ are chain mixing, then D is chain transitive. Moreover, we extended (1)–(4) to three-dimensional cases.

1. Introduction and Preliminaries

Let $t:H\to G$ and $s:G\to H$ be continuous, where $G\subset \mathbb{R}$ and $H\subset \mathbb{R}$ are closed intervals, and let

$$D:G\times H\to G\times H$$

be given by $D(a,b)=\left(t\right(b),s(a\left)\right)$, where $(a,b)\in G\times F$. They have been studied to present a dynamical analysis of the systems which are induced by cournot duopoly games (see [1]). People think that probably the first chaotic concept in a mathematically rigorous way was presented by Li and Yorke [2]. Since then, people gave many different rigorous concepts of chaos. Each of them tries to depict some kind of unpredictability of a system in the future. Akin and Kolyada presented Li–Yorke sensitivity for the first time [3]. Moreover, they presented the notion of spatio-temporal chaos. Schweizer and Smítal defined the concept of distributional chaos [4]. It is well known that the property of distributional chaos is equivalent to the property having positive topological entropy and some other chaotic concepts for some spaces (see [4,5]) and that this kind of equivalence relationship does not hold for the spaces of higher dimensions [6] and zero dimension [7]. In [8], Wang et al. gave the concept of distributional chaos in a sequence and obtained that this kind of chaotic property is equivalent to Li–Yorke chaotic property for continuous maps over a closed interval. In the past few decades, people always paid very close attention to the study of the dynamic behavior of Cournot maps (see [1,9,10,11,12,13]). From [1,12], 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. Let ${\Gamma }_1=\{(t\left(b\right),b):b\in H\}$, ${\Gamma }_2=\{(a,s\left(a\right)):a\in G\}$ and ${\Gamma }_{12}={\Gamma }_1\cup {\Gamma }_2$. Obviously, $D\left({\Gamma }_{12}\right)\subset {\Gamma }_{12}$. ${\Gamma }_{12}$ is called an MPE-set for D (see [9]). At the same time, in [9], Canovas considered a few definitions of chaos for Cournot maps and proved that for each chaotic notion they discussed in [9], D and ${D|}_{{\Gamma }_{12}}$ do not satisfy the same one of them. We know that chaos in Cournot maps has been discussed (see [1,12,13,14,15,16,17]). In [13], Lu and Zhu further explored chaos in Cournot maps. Particularly, they showed that for ${D|}_{{\Gamma }_{12}}$, $D^2{|}_{{\Gamma }_1}$ and $D^2{|}_{{\Gamma }_2}$, they have the same some chaotic properties. Inspired by [9,13], in [18], we further studied the chaotic properties of the above Cournot maps. Particularly, we showed that for a continuous map defined by $D(a,b)=\left(t\right(b),s(a\left)\right)$ over the space $G\times H$, the following statements hold.

(1)

If D satisfies Kato’s concept of chaos, then at least $D^2{|}_{{\Gamma }_1}$ or $D^2{|}_{{\Gamma }_2}$ does.

(2)

Suppose that $D^2{|}_{{\Gamma }_1}$ and $D^2{|}_{{\Gamma }_2}$ have Kato’s chaotic property and that the maps t and s satisfy that for any $\epsilon >0$, if

$$\mid {(s\circ t)}^l\left(v_1\right)-{(s\circ t)}^l\left(v_2\right)\mid <\epsilon$$

and

$$\mid {(t\circ s)}^k\left(u_1\right)-{(t\circ s)}^k\left(u_2\right)\mid <\epsilon$$

for some $l,k\in {\mathbb{Z}}^{+}$ (${\mathbb{Z}}^{+}$ is the set of positive integers), then there is an $m(l,k,\epsilon )\in {\mathbb{Z}}^{+}$ with

$$\mid {(s\circ t)}^{m(l,k,\epsilon )}\left(v_1\right)-{(s\circ t)}^{m(l,k,\epsilon )}\left(v_2\right)\mid <\epsilon ;$$

and

$$\mid {(t\circ s)}^{m(l,k,\epsilon )}\left(u_1\right)-{(t\circ s)}^{m(l,k,\epsilon )}\left(u_2\right)\mid <\epsilon .$$

Then, D has Kato’s chaotic property. In [19], we explored the spatio-temporal chaotic property of a Cournot map $D(a,b)=\left(t\right(b),s(a\left)\right)$ on the space $G\times H$. Particularly, it is established that if both s and t are onto maps, then the followings are equivalent: (1) D has the spatio-temporally chaotic property; (2) $D^2{|}_{{\Gamma }_1}$ has the spatio-temporally chaotic property; (3) $D^2{|}_{{\Gamma }_2}$ has the spatio-temporally chaotic property; and (4) $D^2{|}_{{\Gamma }_1\cup {\Gamma }_2}$ has the spatio-temporally chaotic property. At the same time, it is showed that if s and t are onto maps, then $s\circ t$ has spatio-temporally chaotic property if and only if $t\circ s$ is as well. In [20], by using the Furstenberg family couple, we studied the $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos and the strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaos of a Cournot map $D(a,b)=\left(t\right(b),s(a\left)\right)$ on the product space $G\times H$. In particular, we showed that the following statements hold:

(1)

D having the $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic property (resp. strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic property) is equivalent to $D^2{|}_{{\Lambda }_1}$ having the $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic property (resp. strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic property) and also equivalent to $D^2{|}_{{\Lambda }_2}$ having the $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic property (resp. strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic property).

(2)

D having the $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic property (resp. strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic property) is equivalent to $F^2{|}_{{\Lambda }_1\cup {\Lambda }_2}$ having the $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic property (resp. strong $({\mathcal{F}}_1,{\mathcal{F}}_2)$-chaotic property).

(3)

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

Inspired by [18,19,20] and utilizing the methods in them, we will investigate cofinite sensitivity, shadowing property, P-chaos, and chain mixing in duopoly games in this paper. Let a system

$$D(a,b)=\left(t\right(b),s(a\left)\right)$$

over a product space $G\times H$, where $s:G\to H$, $t:H\to G$, $a\in G$, $b\in H$, and G and H are closed intervals in $\mathbb{R}$. Particularly, it is shown that for any Cournot map $D(a,b)=\left(t\right(b),s(a\left)\right)$ over the space $G\times H$, the following hold:

(1)

D being cofinitely sensitive is equivalent to $D^2{|}_{{\Gamma }_1}$ or $D^2{|}_{{\Gamma }_2}$ being sensitive.

(2)

D possessing an SP is equivalent to $s\circ t$ and $t\circ s$ having the SP.

(3)

$t\circ s$ possesses an SP if and only if $s\circ t$ does as well.

(4)

D is P-chaotic if and only if the maps $s\circ t$ and $t\circ s$ are P-chaotic.

Moreover, the above results can be extended to the three-dimensional case. For chain mixing, we proved the following:

(5)

If D is chain mixing, then both $D^2{|}_{{\Gamma }_1}$ and $D^2{|}_{{\Gamma }_2}$ are chain mixing.

(6)

If $D^2{|}_{{\Gamma }_1}$ and $D^2{|}_{{\Gamma }_2}$ are chain mixing, then D is chain transitive.

Meanwhile, inspired by [9,13,18,20,21], we propose several open questions worth exploring.

Let $(G,d)$ be a metric space. A dynamic system $(G,d,t)$ or a map $t:G\to G$ is called sensitive (see [22,23,24]) if one can find an $\lambda >0$ satisfying that for every nonempty open set $W\subset G:W\ne \varnothing$, one can find two different points $a,b\in W$ satisfying $d(t^k\left(a\right),t^k\left(b\right))>\lambda$ for some integer $k>0$, where $\lambda$ is said to be a sensitivity constant of t. t is cofinitely sensitive (see [25,26]) if there is a $\delta >0$ such that for any nonempty set $A\subset G$, ${\mathbb{Z}}^{+}{N}_t(A,\delta )$ is finite, where

$$N_t(A,\delta )=\{m\in {\mathbb{Z}}^{+}:\mathrm{there}\mathrm{exist}a,b\in A\mathrm{satisfying}|a-b|>\delta \}.$$

Let $t:G\to G$ be a continuous map of a compact metric space $(G,d)$. A sequence $\{u_j:j\ge 0\}\subset G$ is called a $\delta$ pseudo-orbit for t (see [17,25]) if $d(t\left(u_j\right),u_{j+1})<\delta$ for any integer $j\ge 0$. By [21,27], for any given $\epsilon >0$, a given sequence $\{u_j:j\ge 0\}\subset G$ is called to be $\epsilon$-traced by $u\in G$ if $d(t^j\left(u\right),u_j)<\lambda$ for any integer $j\ge 0$. A map $t:G\to G$ is said to possess the POSP if for each $\epsilon >0$, one can find $\delta >0$ satisfying that every $\delta$ pseudo-orbit for t can be $\epsilon$-traced by some point in G.

A dynamic system $(G,d,t)$ or a map $t:G\to G$ having the pseudo-orbit shadowing property (POSP) is said to be P-chaotic if the set $P\left(t\right)$ of all periodic points of t is dense in G (see [22]).

The following lemma is needed and is from [13].

Lemma 1 

(Theorem 3.1 [13]). Assume that $J_1\subset \mathbb{R}$ and $J_2\subset \mathbb{R}$ are two compact subintervals and $g_j:J_j\to J_j$ is continuous for any $j\in \{1,2\}$. Then, $g_1\times g_2$ is $\mathcal{Q}$-chaotic is equivalent to that so is g, where $g\in \{g_1,g_2\}$ is $\mathcal{Q}$-chaotic, where $\mathcal{Q}$-chaoticity is one of the following chaotic properties: Li–Yorke sensitivity, Li–Yorke chaos, distributional chaos, distributional chaos in a sequence.

2. Cofinite Sensitivity

In [13], the authors show that for any Cournot map D, D has the Li–Yorke sensitive property that is equivalent to that one of the following holding.

(1)

$D^2{|}_{{\Gamma }_1}$ has the Li–Yorke sensitive property.

(2)

$D^2{|}_{{\Gamma }_2}$ has the Li–Yorke sensitive property.

Inspired by the above result, we obtain the following results.

Theorem 1. 

Assume that the metric d over the space $G\times H$ is defined as

$$d((a_1,b_1),(a_2,b_2))=max\left\{\right|a_2-a_1|,|b_2-b_1\left|\right\}$$

and the product map $s\times t$ of $s:G\to G$ and $t:H\to H$ is defined as

$$(s\times t)(a,b)=\left(s\right(a),t(b\left)\right)$$

for any $a\in G$ and any $b\in H$ where $G\subset \mathbb{R}$ and $H\subset \mathbb{R}$ are compact subintervals, and let $D(a,b)=\left(t\right(b),s(a\left)\right)$ be a Cournot map. Then, D being cofinitely sensitive is equivalent to one of the following conditions being satisfied.

(1) 

$D^2{|}_{{\Gamma }_1}$ is sensitive.

(2) 

$D^2{|}_{{\Gamma }_2}$ is sensitive.

Proof. 

Let D be cofinitely sensitive. By Theorem 31 in [25], $D^2$ is cofinitely sensitive, which implies that $D^2$ is sensitive. Since

$$D^2=(t\circ s)\times (s\circ t),$$

According to Theorem 3.1 of [13], one can see that $t\circ s$ or $s\circ t$ is sensitive. Using Theorem 2 in [21], $t\circ s$ or $s\circ t$ is cofinitely sensitive. Without any loss of generality, it is assumed that $t\circ s$ is cofinitely sensitive. According to the definition, one can find a constant ${\delta }^{\prime }>0$ of $t\circ s$ with ${\delta }^{\prime }<\delta$ satisfying that for every $a\in G$ and any $\eta >0$, one can find an $b_{\eta }\in G$ with $|a-b_{\eta }|<\eta$ and

$${|(t\circ s)}^n\left(a\right)-{(t\circ s)}^n\left(b_{\eta }\right)|>{\delta }^{\prime }$$

for any integer $n\ge m$ some integer $m>0$. Since s is uniformly continuous, one can find an ${\eta }_1\in (0,\eta )$ with

$$|s\left(a_1\right)-s\left(a_2\right)|<\eta$$

for any $a_1,a_2\in G$ with

$$|a_1-a_2|<{\eta }_1.$$

Clearly,

$$d((a,s\left(a\right)),(b_{{\eta }_1},s\left(b_{{\eta }_1}\right)))<\eta .$$

Hence,

$$d(D^{2n}(a,s\left(a\right)),D^{2n}(b_{{\eta }_1},s\left(b_{{\eta }_1}\right)))\ge |{(t\circ s)}^n\left(a\right)-{(t\circ s)}^n\left(b_{{\eta }_1}\right)|>{\delta }^{\prime }$$

for any integer $n\ge m$. This shows that $D^2{|}_{{\Gamma }_2}$ is cofinitely sensitive.

Now, we suppose that $D^2{|}_{{\Gamma }_2}$ is cofinitely sensitive having a sensitivity constant $\delta >0$. According to the uniform continuity of s, for the above $\delta >0$ and any $a_1,a_2\in G$ with $a_1\ne a_2$, one can find some ${\delta }^{\prime }>0$ with ${\delta }^{\prime }<\delta$ such that

$$|a_1-a_2|\le {\delta }^{\prime }$$

implies

$$|s\left(a_1\right)-s\left(a_2\right)|\le \delta .$$

Consequently, for any $a_1,a_2\in G$ with $a_1\ne a_2$ and

$$|a_1-a_2|\le {\delta }^{\prime },$$

if

$$d(D^{2n}(a_1,s\left(a_1\right)),a_2,s\left(a_2\right)))>\delta$$

for any integer $n\ge m$ and some integer $m>0$, then

$${|(t\circ s)}^n\left(a_1\right)-{(t\circ s)}^n\left(a_2\right)\left)\right|>{\delta }^{\prime }.$$

This means that $t\circ s$ is cofinitely sensitive. Since

$$D^2=(t\circ s)\times (s\circ t),$$

by Lemma 3.3 from [25], $D^2$ is cofinitely sensitive. According to Theorem 31 of [25], D is cofinitely sensitive. □

Remark 1. 

In Theorem 1, we improve the result of Theorem 2.2 in [18].

Due to the fact that composite mappings satisfy the law of union, when one of the mappings in $s\circ \phi \circ t$ is an identity mapping, $s\circ \phi \circ t$ becomes $s\circ t$, $\phi \circ t$, or $s\circ \phi$, which is the case in two dimensions. Therefore, it is natural to consider whether the previous results in Section 3 can be extended to three dimensions. Moreover, the n-dimensional situation is similar.

Let $G,F,H$ be compact subintervals of $\mathbb{R}$, and let $t:G\to F$, $\phi :F\to H$, $s:H\to G$ be continuous maps. Define $\tilde{D}(a,b,c)=(s\left(c\right),t\left(b\right),\phi \left(a\right))$, where $(a,b,c)\in G\times F\times H$. And

$$d((a_1,b_1,c_1),(a_2,b_2,c_2))=max\left\{\right|a_2-a_1|,|b_2-b_1|,|c_2-c_1\left|\right\}.$$

Denote

$${\tilde{\Gamma }}_1=\{a,t\left(a\right),\phi \circ t\left(a\right):a\in G\};$$

$${\tilde{\Gamma }}_2=\{s\circ \phi \left(b\right),b,\phi \left(b\right):b\in F\};$$

$${\tilde{\Gamma }}_3=\left\{\right\}s\left(c\right),t\circ s\left(c\right),\phi \circ t\circ s\left(c\right):c\in H\}.$$

One can obtain that

$${\tilde{D}}^3=(s\circ \phi \circ t)\times (t\circ s\circ \phi )\times (\phi \circ t\circ s),$$

then, we have conclusions similar to Theorem 1.

Theorem 2. 

$\tilde{D}$ being cofinitely sensitive is equivalent to one of the following conditions being satisfied.

(1) 

${\tilde{D}}^3{|}_{{\tilde{\Gamma }}_1}$ is sensitive;

(2) 

${\tilde{D}}^3{|}_{{\tilde{\Gamma }}_2}$ is sensitive;

(3) 

${\tilde{D}}^3{|}_{{\tilde{\Gamma }}_3}$ is sensitive.

Proof. 

Let $\tilde{D}$ be cofinitely sensitive. According to Theorem 31 in [25], ${\tilde{D}}^3$ is cofinitely sensitive. Then, ${\tilde{D}}^3$ is sensitive. Since ${\tilde{D}}^3=(s\circ \phi \circ t)\times (t\circ s\circ \phi )\times (\phi \circ t\circ s)$, by Theorem 3.1 in [13], $s\circ \phi \circ t$, $t\circ s\circ \phi$, or $\phi \circ t\circ s)$ is sensitive. So, by Theorem 2 in [21], $s\circ \phi \circ t$, $t\circ s\circ \phi$, or $\phi \circ t\circ s)$ is cofinitely sensitive.

Without loss of generality, let $s\circ \phi \circ t$ be cofinitely sensitive. Then, one can find a constant ${\delta }^{*}:0<{\delta }^{*}<\delta$, where for every $a\in G$ and any $\eta >0$, there exists a $b_{\eta }\in G:|a-b_{\eta }|<\eta$ such that

$${|(s\circ \phi \circ t)}^n\left(a\right)-{(s\circ \phi \circ t)}^n\left(b_{\eta }\right)|>{\delta }^{*}$$

for any integer $n\ge m$ and some integer $m>0$. Since t and $\phi \circ t$ are uniformly continuous, then one can find two points ${\eta }_1,{\eta }_2\in (0,\eta )$ such that

$$|t\left(a_1\right)-t\left(a_2\right)|<\eta ,|(\phi \circ t)\left(a_1\right)-(\phi \circ t)\left(a_2\right)|<{\eta }_2,$$

where $a_1,a_2\in G$ and $|a_1-a_2|<\eta$. So,

$$d[(a,t\left(a\right),(\phi \circ t)\left(a\right)),(b_{{\eta }_1},t\left(b_{{\eta }_1}\right),(\phi \circ t)\left(b_{{\eta }_1}\right))]<\eta .$$

Hence,

$$d[{\tilde{D}}^{3n}(a,t\left(a\right),(\phi \circ t)\left(a\right)),{\tilde{D}}^{3n}(b_{{\eta }_1},t\left(b_{{\eta }_1}\right),(\phi \circ t)\left(b_{{\eta }_1}\right))]\ge |(t\circ \phi \circ s)\left(a\right)-(t\circ \phi \circ s)\left(b_{{\eta }_1}\right)|>{\delta }^{*}$$

for any integer $n\ge m$. This implies that ${\tilde{D}}^3{|}_{\tilde{\Gamma }1}$ is cofinitely sensitive.

Now, let ${\tilde{D}}^3{|}_{\tilde{\Gamma }1}$ be cofinitely sensitive, and the sensitive constant is $\delta >0$. Since t and $\phi \circ t$ are uniformly continuous, if the sensitive constant is $\delta >0$ and any $a_1,a_2\in G:a_1\ne a_2$, one can find a ${\delta }^{\prime }:0<{\delta }^{\prime }<\delta$ such that $|a_1-a_2|\le {\delta }^{\prime }$ implies $|t\left(a_1\right)-t\left(a_2\right)|\le \delta$ and $|(\phi \circ t)\left(a_1\right)-(\phi \circ t)\left(a_2\right)|\le \delta .$ So, for any $a_1,a_2\in G:0<|a_1-a_2|\le {\delta }^{\prime }$, if $d[{\tilde{D}}^3(a_1,t\left(a_1\right),(\phi \circ t)\left(a_1\right),(a_2,t\left(a_2\right),(\phi \circ t)\left(a_2\right))]>\delta$ for some $m>0$ and any integer $n:n\ge m$, then

$${|(s\circ \phi \circ t)}^n\left(a_1\right)-{(s\circ \phi \circ t)}^n\left(a_2\right)|>{\delta }^{\prime }.$$

This implies that $s\circ \phi \circ t$ is cofinitely sensitive.

Since ${\tilde{D}}^3=(s\circ \phi \circ t)\times (t\circ s\circ \phi )\times (\phi \circ t\circ s)$, by Lemma 3.3 in [25], ${\tilde{D}}^3$ is cofinitely sensitive. Then, by Theorem 31 in [25], $\tilde{D}$ is cofinitely sensitive. □

3. Shadowing Property (SP) and P-Chaoticity

Theorem 3. 

Assume that $D(a,b)=\left(t\right(b),s(a\left)\right)$ is a Cournot map. Then, D has an SP that is equivalent to the following being true:

(1) 

$s\circ t$ possesses the SP.

(2) 

$t\circ s$ possesses the SP.

Proof. 

It is known that $p\times q$ has an SP that is equivalent to both p and q possessing the SP, and that $p^2$ has an SP equivalent to p possessing the SP. By

$$D^2=(t\circ s)\times (s\circ t),$$

we complete the proof of this theorem. □

Theorem 4. 

Assume that $D(a,b)=\left(t\right(b),s(a\left)\right)$ is a Cournot map. Then, D having dense periodic points is equivalent to both $s\circ t$ and $t\circ s$ possessing dense periodic points.

Proof. 

Let $P\left(g\right)$ be the set of all periodic points of a map g over a space X. Clearly,

$$P\left(g^2\right)=P\left(g\right)$$

and

$$P(g\times f)=P\left(g\right)\times P\left(f\right),$$

where f is a map of a space Y. By

$$D^2=(t\circ s)\times (s\circ t),$$

we complete the proof of this theorem. □

Theorem 5. 

Assume that $D(a,b)=\left(t\right(b),s(a\left)\right)$ is a Cournot map. If both $D^2{|}_{{\Gamma }_1}$ and $D^2{|}_{{\Gamma }_2}$ possess dense periodic points, then D possesses dense periodic points.

Proof. 

Clearly, $P\left(D^2{|}_{{\Gamma }_1}\right)\cup P\left(D^2{|}_{{\Gamma }_2}\right)\subset P\left(D^2\right)$. Then, both $D^2{|}_{{\Gamma }_1}$ and $D^2{|}_{{\Gamma }_2}$ possessing dense periodic points’ implies that $D^2$ possess dense periodic points. By $P\left(D^2\right)=P\left(D\right)$, D possess dense periodic points. □

Question 1. 

Assume that $D(a,b)=\left(t\right(b),s(a\left)\right)$ has dense periodic points. Does $D^2{|}_{{\Gamma }_i}(i=1,2)$ possess dense periodic points?

Theorem 6. 

Assume that $D(a,b)=\left(t\right(b),s(a\left)\right)$ is a Cournot map. Then, $t\circ s$ possessing an SP is equivalent to $s\circ t$ possessing it as well.

Proof. 

Assume that $s\circ t$ possesses the SP. Let $\eta >0$. Since t is uniformly continuous, for the above $\eta >0$, one can find an ${\eta }^{\prime }>0$ with ${\eta }^{\prime }<{\displaystyle \frac{1}{2}}\eta$ satisfying the following: if $b_1,b_2\in H$ and $|b_1-b_2|<{\eta }^{\prime }$, then

$$|t\left(b_1\right)-t\left(b_2\right)|<{\displaystyle \frac{1}{2}}\eta .$$

As $s\circ t$ possesses the SP, by the definition, there is a ${\delta }^{\prime }>0$ with ${\delta }^{\prime }\le {\eta }^{\prime }$ such that any ${\delta }^{\prime }$ pseudo-orbit for $s\circ t$ is ${\eta }^{\prime }$-traced by some point of H. Since s is uniformly continuous, for the above ${\delta }^{\prime }>0$, there is a $\delta >0$ with $\delta <{\delta }^{\prime }$ satisfying if $a_1,a_2\in G$ and $|a_1-a_2|<\delta$ then

$$|s\left(a_1\right)-s\left(a_2\right)|<{\delta }^{\prime }.$$

Let

$${\left\{a_k\right\}}_{k\ge 0}\subset G$$

be a $\delta$ pseudo-orbit for $t\circ s$. Then,

$${\left\{s\left(a_k\right)\right\}}_{k\ge 0}\subset H$$

is a ${\delta }^{\prime }$ pseudo-orbit for $s\circ t$. By the definition, there is a $b\in H$ satisfying that

$${|(s\circ t)}^k\left(b\right)-s\left(a_k\right)|<{\eta }^{\prime }$$

for any $k\ge 0$. So,

$${|(t\circ s)}^k\left(t\left(b\right)\right)-(t\circ s)\left(a_k\right)|<{\displaystyle \frac{1}{2}}\eta$$

for any $k\ge 0$. Consequently,

$${|(t\circ s)}^{k-1}(t\circ s\circ t)\left(b\right)-a_{k-1}|$$

$${\le |(t\circ s)}^k\left(t\left(b\right)\right)-(t\circ s)\left(a_k\right)|+|(t\circ s)\left(a_k\right)-a_{k-1}|<{\displaystyle \frac{1}{2}}\eta +\delta <\eta$$

for any $k\ge 1$. By the definition, $t\circ s$ has the SP. Similarly, it is easy to deduce that if $t\circ s$ possesses the SP, then so does $s\circ t$. □

Question 2. 

Let $D,D^2,{\Gamma }_1,{\Gamma }_2$ be defined as above.

(1) 

If $D^2{|}_{{\Gamma }_1}$ and $D^2{|}_{{\Gamma }_2}$ both have the SP, does D also have the SP or vice versa?

(2) 

If $D^2{|}_{{\Gamma }_1}$ has the SP, does $D^2{|}_{{\Gamma }_2}$ also have the SP or vice versa?

Question 3. 

Let $D,D^2,{\Gamma }_1,{\Gamma }_2$ be defined as above.

(1) 

If $D^2{|}_{{\Gamma }_1}$ and $D^2{|}_{{\Gamma }_2}$ both are P-chaotic, is D also P-chaotic or vice versa?

(2) 

If $D^2{|}_{{\Gamma }_1}$ is P-chaotic, is $D^2{|}_{{\Gamma }_2}$ also P-chaotic or vice versa?

Theorem 7. 

Let $D(a,b)=\left(t\right(b),s(a\left)\right)$ be a Cournot map. Then, D having the P-chaotic property is equivalent to the maps $s\circ t$ and $t\circ s$ having the corresponding property.

Proof. 

According to the definition and Theorems 3 and 4, Theorem 7 is true. □

Theorem 8. 

$\tilde{D}$ having the SP is equivalent to the following being true.

(1) 

$s\circ \phi \circ t$ possesses the SP;

(2) 

$t\circ s\circ \phi$ possesses the SP;

(3) 

$\phi \circ t\circ s$ possesses the SP.

Proof. 

It is easy to obtain that $(s\circ \phi \circ t)\times (t\circ s\circ \phi )\times (\phi \circ t\circ s)$ possessing the SP is equivalent to $s\circ \phi \circ t$, $t\circ s\circ \phi$, and $\phi \circ t\circ s$ all possessing the SP. Since ${\tilde{D}}^3=(s\circ \phi \circ t)\times (t\circ s\circ \phi )\times (\phi \circ t\circ s)$, then conclusions are established. □

Theorem 9. 

$\tilde{D}$ having dense periodic points is equivalent to $s\circ \phi \circ t$, $t\circ s\circ \phi$, and $\phi \circ t\circ s$ all possessing dense periodic points.

Proof. 

It is easy to obtain that $P({\tilde{D}}^3{|}_{\tilde{\Gamma }1})\cup P({\tilde{D}}^3{|}_{\tilde{\Gamma }2})\cup P({\tilde{D}}^3{|}_{\tilde{\Gamma }3})\subset P({\tilde{D}}^3)$, where $P(·)$ denotes the set of periods of a mapping. Then, if ${\tilde{D}}^3{|}_{\tilde{\Gamma }i}(i=1,2,3)$ has dense periodic points, then ${\tilde{D}}^3$ also has this property. And because $P({\tilde{D}}^3)=P(\tilde{D})$, then $s\circ \phi \circ t$, $t\circ s\circ \phi$, and $\phi \circ t\circ s$ all possess dense periodic points. □

Theorem 10. 

$s\circ \phi \circ t$ possessing the SP is equivalent to $t\circ s\circ \phi$ possessing the SP as well as $\phi \circ t\circ s$ possessing the SP.

Proof. 

Since $s\circ \phi \circ t=(s\circ \phi )\circ t$, $t\circ s\circ \phi =t\circ (s\circ \phi )$, and by Theorem 6, $(s\circ \phi )\circ t$ possessing the SP is equivalent to $t\circ (s\circ \phi )$ possessing the SP. So, $s\circ \phi \circ t$ possessing the SP is equivalent to $t\circ s\circ \phi$ possessing the SP.

Similarly, since $s\circ \phi \circ t=s\circ (\phi \circ t)$, $\phi \circ t\circ s=(\phi \circ t)\circ s$, and by Theorem 6, $s\circ (\phi \circ t)$ possessing the SP is equivalent to $(\phi \circ t)\circ s$ possessing the SP. So, $s\circ \phi \circ t$ possessing the SP is equivalent to $\phi \circ t\circ s$ possessing the SP.

This complete the proof. □

Theorem 11. 

$\tilde{D}$ having the P-chaotic property is equivalent to the maps $s\circ \phi \circ t$$t\circ s\circ \phi$, and $\phi \circ t\circ s$ having the corresponding property.

Proof. 

According to the definition of P-chaos and Theorems 8 and 9, Theorem 11 is true. □

Example 1. 

Let $f\left(a\right)=1-|1-2a|,a\in [0,1]$, and $s\left(a\right)=\phi \left(a\right)=t\left(a\right)=f\left(a\right)$ for any $a\in [0,1]$; then, Theorems 1–11 hold.

4. Chain Mixing

Let $(G,s)$ be a dynamical system on a metric space $(G,d)$. For $x,y\in G$, an $\epsilon$-chain from x to y of length m is a finite sequence $x_0=x,x_1,\cdots ,x_m=y$ such that $d(s\left(x_j\right),x_{j+1})\le \epsilon$ for $j=0,1,\cdots \cdots ,m-1$. The system $(G,s)$ or the map s is said to be chain transitive if for any $\epsilon >0$ and any two points $x,y\in G$, there is an $\epsilon$-chain from x to y. The system $(G,s)$ or the map s is said to be $\epsilon$-chain mixing if there is some $N>0$ such that for any $x,y\in G$ and any integer $m\ge N$, there is an $\epsilon$-chain from x to y such that its length is m. The system $(G,s)$ or the map s is said to be chain mixing if it is $\epsilon$-chain mixing for any $\epsilon >0$.

Theorem 12. 

Assume that t and s are surjective. Then, $t\circ s$ is chain mixing if and only if $s\circ t$ is as well.

Proof. 

Assume that $s\circ t$ is chain mixing. Let $\epsilon >0$. Since t is uniformly continuous, for the above $\epsilon >0$, there is ${\epsilon }^{\prime }>0$ with ${\epsilon }^{\prime }<{\displaystyle \frac{1}{2}}\epsilon$ such that if $b_1,b_2\in H$ and $|b_1-b_2|<{\epsilon }^{\prime }$, then

$$|t\left(b_1\right)-t\left(b_2\right)|<\epsilon .$$

As $s\circ t$ is chain mixing, by the definition, there is an integer $N>0$ such that for any integer $n>N$ and any $a,b\in H$, one has an ${\epsilon }^{\prime }$-chain of length n from a to b; that is, $a_0,a_1,\cdots ,a_n\in H$ satisfy that $a_0=a,a_n=b$, and

$$|(s\circ t)\left(a_j\right)-a_{j+1}|<{\epsilon }^{\prime }$$

for any

$$j\in \{0,1,\cdots ,n-1\}.$$

This implies that $t\left(a_0\right),t\left(a_1\right),\cdots ,t\left(a_n\right)\in t\left(H\right)$ such that $t\left(a_0\right)=t\left(a\right),t\left(a_n\right)=t\left(b\right)$, and

$$|t\circ (s\circ t)\left(a_j\right)-t\left(a_{j+1}\right)|<\epsilon$$

for any

$$j\in \{0,1,\cdots ,n-1\}.$$

Since t is surjective, by the definition and the above argument, $t\circ s$ is chain mixing. By a similar argument, one can deduce that if $t\circ s$ is chain mixing, then so is $s\circ t$. □

Theorem 13. 

Assume that $D(a,b)=\left(t\right(b),s(a\left)\right)$ is a Cournot map, t and s are surjective. Then, the following two statements are true:

(1) 

If D is chain mixing, then both $D^2{|}_{{\Gamma }_1}$ and $D^2{|}_{{\Gamma }_2}$ are chain mixing.

(2) 

If $D^2{|}_{{\Gamma }_1}$ and $D^2{|}_{{\Gamma }_2}$ are chain mixing, then D is chain transitive.

Proof. 

Assume that D is chain mixing. By Corollary 12 in [28], $D^2$ is chain mixing. By $D^2=(t\circ s)\times (s\circ t)$, the chain mixing of $D^2$ and the definition, $t\circ s$ and $s\circ t$ is chain mixing. Suppose that $t\circ s$ is chain mixing. By the hypothesis and Theorem 3.8, $s\circ t$ is chain mixing. By the definition, one can easily verify that $D^2=(t\circ s)\times (s\circ t)$ are chain mixing. Let $\epsilon >0$. As s is uniformly continuous, for the above $\epsilon >0$, there is ${\epsilon }_1>0$ with ${\epsilon }_1<\epsilon$ such that if

$$|a_1-a_2|<{\epsilon }_1$$

and $a_1,a_2\in G$, then

$$|s\left(a_1\right)-s\left(a_2\right)|<\epsilon .$$

As $t\circ s$ is chain mixing, by the definition, for the above ${\epsilon }_1>0$, there is an integer $N>0$ such that for any integer $n>N$ and any $a,b\in G$, there is an ${\epsilon }_1$-chain of length n from a to b; that is, there are $a_0,a_1,\cdots ,a_n\in G$ which satisfy that $a_0=a,a_n=b$, and

$$|(t\circ s)\left(a_j\right)-a_{j+1}|<{\epsilon }^{\prime }$$

for any

$$j\in \{0,1,\cdots ,n-1\}.$$

By the hypothesis, $s\left(a_0\right),s\left(a_1\right),\cdots ,s\left(a_n\right)\in s\left(G\right)$ such that $s\left(a_0\right)=s\left(a\right),s\left(a_n\right)=s\left(b\right)$, and

$$|s\circ (t\circ s)\left(a_j\right)-s\left(a_{j+1}\right)|<\epsilon$$

for any

$$j\in \{0,1,\cdots ,n-1\}.$$

Then, we have

$$d(D^2\left((a_j,s\left(a_j\right))\right),(a_{j+1},s\left(a_{j+1}\right)))<\epsilon$$

for every integer $j\in \{0,1,\cdots ,n-1\}$. As s is surjective, by the hypothesis and the above argument, $D^2{|}_{{\Gamma }_2}$ is chain mixing. Similarly, one can show that if $s\circ t$ is chain mixing, then so is $D^2{|}_{{\Gamma }_1}$.

Now, assume that $D^2{|}_{{\Gamma }_i}$ is chain mixing for every $i\in \{1,2\}$, and let $\epsilon >0$. By the definition, there is an integer $N>0$ such that for any integer $n>N$ and any $a,b\in H$ (resp. $a^{\prime },b^{\prime }\in G$), there is an $\epsilon$-chain of length n from $\left(t\right(a),a)$ to $\left(t\right(b),b)$ (resp. from $(a^{\prime },s\left(a^{\prime }\right))$ to $(b^{\prime },s\left(b^{\prime }\right))$); that is, there are $(t\left(a_0\right),a_0),(t\left(a_1\right),a_1),\cdots ,(t\left(a_n\right),a_n)\in {\Gamma }_1$ (resp. $(a_0^{\prime },s\left(a_0^{\prime }\right)),(a_1^{\prime },s\left(a_1^{\prime }\right)),\cdots ,(a_n^{\prime },s\left(a_n^{\prime }\right))\in {\Gamma }_2$) which satisfy that $(t\left(a_0\right),a_0)=(t\left(a\right),a),(t\left(a_n\right),a_n)=(t\left(b\right),b)$, $|(t\circ s)\left(t\left(a_j\right)\right)-t\left(a_{j+1}\right)|<\epsilon$ and $|(s\circ t)\left(a_j\right)-a_{j+1}|<\epsilon$ (resp. $(a_0^{\prime },s\left(a_0^{\prime }\right))=(a^{\prime },s\left(a^{\prime }\right)),(a_n^{\prime },s\left(a_n^{\prime }\right))=(b^{\prime },s\left(b^{\prime }\right))$, $|(t\circ s)\left(a_j^{\prime }\right)-a_{j+1}^{\prime }|<\epsilon$ and $|(s\circ t)\left(s\left(a_j^{\prime }\right)\right)-s\left(a_{j+1}^{\prime }\right)|<\epsilon$) for any $j\in \{0,1,\cdots ,n-1\}$. Then, an $\epsilon$-chain

$$(a^{\prime },a)=(a_0^{\prime },a_0),(t\left(a_0\right),s\left(a_0^{\prime }\right)),(a_1^{\prime },a_1),(t\left(a_1\right),s\left(a_1^{\prime }\right)),(a_2^{\prime },a_2),\cdots ,(a_n^{\prime },a_n)=(b^{\prime },b)$$

of length $2n+1$ from $(a^{\prime },a)$ to $(b^{\prime },b)$ is obtained. By the hypothesis and the definition, D is chain transitive. □

Question 4. 

We do not know if the following are true:

(1) 

If D is chain transitive, is $D^2{|}_{{\Gamma }_i}$ chain transitive for every $i\in \{1,2\}$?

(2) 

If D is chain mixing, is $D^2{|}_{{\Gamma }_i}$ chain mixing for every $i\in \{1,2\}$?

5. Conclusions

In this research, we studied the sensitivity, shadowing property, P-chaoticity, and chain mixing of duopoly games. Necessary and sufficient conditions for cofinite sensitivity or P-chaoticity or SP of D are obtained. And these conclusions can be extended to a three-dimensional case. Moreover, sufficient conditions for D having dense periodic points or D being chain mixing are given, and some unresolved questions have been raised. In the future, we will continue to study these issues.