# Chaos in Duopoly Games via Furstenberg Family Couple

Authors: Yu Zhao; Risong Li
Journal: Complexity
 (2019)

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## Abstract

Assume thatH1 and H2 are two given closed subintervals of ℝ and that f2:H1⟶H2 and f1:H2⟶H1 are continuous maps. Let ϒh1,h2=f1h2,f2h1 be a Cournot map over the space H1×H2. In this paper, we study G1,G2-chaos (resp. strong G1,G2-chaos) of such a Cournot map. We will show that the following are true: (1) ϒ is G1,G2-chaotic (resp. strong G1,G2-chaotic) if and only if ϒ2Λ1 is G1,G2-chaotic (resp. strong G1,G2-chaotic) if and only if Γ2Λ2 is G1,G2-chaotic (resp. strong G1,G2-chaotic). (2) ϒ is G1,G2-chaotic (resp. strong G1,G2-chaotic) if and only if ϒ2Λ1∪Λ2 is G1,G2-chaotic (resp. strong G1,G2-chaotic). (3) f1∘f2 is G1,G2-chaotic (resp. strong G1,G2-chaotic) if and only if so is f2∘f1. MR(2000) Subject Classification: Primary 37D45, 54H20, and 37B40 and Secondary 26A18 and 28D20.


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## Body

## 1. Introduction

LetH1 and H2 be closed subintervals of ℝ, and let f1:H2⟶H1 and f2:H1⟶H2 be continuous. In the whole paper, ϒ:H1×H2⟶H1×H2 is defined by ϒh1,h2=f1h2,f2h1 for any h1,h2∈H1×H2. Such a map has been investigated to give a mathematical analysis of Cournot duopoly (see [1]). Probably the first notion of chaos in a mathematically rigorous way was posed by Li and Yorke [2]. Since then, a lot of different notions of chaos have been posed. Akin and Kolyada gave the concept of Li–Yorke sensitivity for the first time [3]. They also gave the concept of spatiotemporal chaos. Schweizer and Smítal gave the concept of distributional chaos [4]. We know that distributional chaos is equivalent to positive topological entropy and some other chaotic properties for some particular spaces (see [4, 5]), and that this equivalence relationship will become invalid for some higher dimensional spaces [6] and some zero-dimensional spaces [7]. In [8], Wang et al. gave the definition of distributional chaos with respect to a sequence and got that such chaos is equivalent to Li–Yorke chaos for continuous maps over a closed subinterval. Over the past few decades, people have been paying very close attention to the chaotic properties of Cournot maps (see [1,9–13]). From [1, 12] one can see that there exist Markov perfect equilibria processes. That is, two fixed players move alternatively and ensure that any of them chooses the best reply to the previous action of another player. Put Λ1=f1h2,h2:h2∈H2, Λ2=h1,f2h1:h1∈H1, and Λ12=Λ1∪Λ2. Obviously, ϒQ12⊂Λ12. The set Λ12 is said to be a MPE set for ϒ (see [9]). Moreover, in [9], the authors studied several kinds of chaos for Cournot maps and obtained that for any definition they considered in [9], and it does not satisfy the condition that ϒ is chaotic if and only if so is ϒΛ12. It is well known that some chaotic properties of Cournot maps have been explored (see [1,12–17]). Recently, Lu and Zhu further studied some chaotic properties of Cournot maps and showed that some chaotic properties of ϒΛ12, ϒ2Λ1 and ϒ2Λ2 are same. In this paper, it is shown that for any Cournot map ϒh1,h2=f1h2,f2h1 over the product space H1×H2, the following properties are hold:(1)
ϒ is G1,G2-chaotic (resp. strong G1,G2-chaotic) if and only if ϒ2Λ1 is G1,G2-chaotic (resp. strong G1,G2-chaotic) if and only if Γ2Λ2 is G1,G2-chaotic (resp. strong G1,G2-chaotic).(2)
ϒ is G1,G2-chaotic (resp. strong G1,G2-chaotic) if and only if ϒ2Λ1∪Λ2 is G1,G2-chaotic (resp. strong G1,G2-chaotic)(3)
f1∘f2 is G1,G2-chaotic (resp. strong G1,G2-chaotic) if and only if so is f2∘f1


## 2. Preliminaries

LetH,ξ be a compact metric space. A dynamic system H,f means that f is a continuous self-map over the space H.Letf:H⟶H be a map on the space H,ξ. The map f is chaotic in the sense of Li–Yorke if there is an uncountable set C⊂H satisfying that for any h1,h2∈C with h1≠h2:(1)liminfl⟶∞ξflh1,flh2=0,limsupl⟶∞ξflh1,flh2>0.This uncountable setC is called a scrambled set of f.An important generalization of Li–Yorke chaos is distributional chaos, which is given in 1994 by Puu and Sushko [1].LetH,ξ be a metric space and f:H⟶H be continuous. For any h1,h2∈H, the upper (lower) distribution function Fh1h2∗t,f (Fh1h2t,f) deduced by h1,h2 and f is defined by(2)Fh1h2∗t,f=limsupm⟶∞1m∑j=1mχ0,tξfjh1,fjh2,Fh1h2t,f=liminfm⟶∞1m∑j=1mχ0,tξfjh1,fjh2,where χ0,t is the characteristic function of the set 0,t. The map f is distributional chaotic if there is an uncountable subset C⊂H satisfying that for any h1≠h2∈C, Fh1h2∗t,f=1∀t>0, and ∃t>0 such that Fh1h2t,f=0. This uncountable subset C is called a distributionally scrambled set of f. And this point pair h1,h2 which satisfies the above two conditions is called a distributionally scrambled pair of f.In 1997, Furstenberg family is introduced by Akin [18]. Then, Xiong and Tan defined G1,G2-chaos and described chaos via Furstenberg family couple. Also, they obtained some sufficient conditions of G1,G2-chaos (see [19]).Letℕ=1,2,3,…, ℕ0=ℕ∪0, and P be the collection of all subsets of ℕ0. A collection G⊂P is called a Furstenberg family (see [19]) if it satisfies that if G1⊂G2 and G1∈G then G2∈G. A family G is said to be proper if it is a proper subset of P (see [19]). In the whole paper, we suppose that all Furstenberg families are proper. Clearly, a family G is proper if and only if ℕ0∈G and ∅∉G (see [19]).For any Furstenberg familiesG1 and G2 and any map f:H⟶H, C⊂H is called a G1,G2-scrambled set of f (see [19]), if ∀h1≠h2∈C, the following two conditions are satisfied:(1)
∀t>0, m∈ℕ:ξfmh1,fmh2<t∈G1(2)
∃δ>0, m∈ℕ:ξfmh1,fmh2>δ∈G2This pairh1,h2 is called a G1,G2-scrambled pair of f. The map f is said to be G1,G2-chaotic if there is an uncountable G1,G2-scrambled set of f. When G1=G2=G, the map f is said to be G-chaotic and the pair h1,h2 is a G-scrambled pair. The map f is said to be strong G1,G2-chaotic if one can find δ>0 satisfying that for any h1≠h2∈C, ∀t>0, m∈ℕ:ξfmh1,fmh2<t∈G1 and m∈ℕ:ξfmh1,fmh2>δ∈G2.Similarly, one can give the concept of strongG-chaos.LetG⊂ℕ. The upper density μ¯G and the lower density μ¯G of G are defined by(3)μ¯G=lim supm⟶∞#G∩1,2,…,mm,μ¯G=lim infm⟶∞#G∩1,2,…,mm,where #G denotes the cardinality of the set G.Letℬ denotes the set of all infinite subsets of ℕ. For arbitrary t∈0,1, put M¯t=F∈ℬ:μ¯F≥t. Then, a pair h1,h2 is a M¯0,M¯0-scrambled pair if and only if it is a Li–Yorke scrambled pair (see [19]). A pair h1,h2 is a M¯1,M¯1-scrambled pair if and only if it is a distributionally scrambled pair (see [19]). Hence, M¯0,M¯0-chaos is Li–Yorke chaos, and M¯1,M¯1-chaos is distributional chaos.Fori∈ℕ0 and G∈P, let G+i=j+i:j∈G∩ℕ0 and G−i=j−i:j∈G∩ℕ0. A Furstenberg family G is said to be translation-invariant if for any G∈G and any i∈ℕ0 G+i∈G and G−i∈G. It is easily seen that ℬ is a proper and translation-invariant family (see [19]).Clearly, for anyt∈0,1, M¯t is a translation-invariant Furstenberg family and M¯0 = ℬ (see [19]).


## 3. Main Results

Theorem 1.
Let the product metricξ on the product space H1×H2 be defined by ξa1,b1,a2,b2=maxa2−a1,b2−b1 and the product map π1×π2 of π1:H1⟶H1 and π2:H2⟶H2 be defined by π1×π2a,b=π1a,π2b for any a∈H1 and any b∈H2, where H1,H2⊂ℝ are compact intervals, and let ϒa,b=f1b,f2a be a Cournot map. If G1 and G2 are two Furstenberg families such that G2 is translation-invariant, then f1∘f2 is G1,G2-chaotic if and only if so is f2∘f1.Proof.
Suppose thatf1∘f2 is G1,G2-chaotic. By the definition, there is an uncountable G1,G2-scrambled set D⊂H1 of f1∘f2. By the definition, for any given b>0 and any h1,h2∈H1 with h1≠h2 one has that(4)m∈ℕ:f1∘f2mh1−f1∘f2mh2<b∈G1.
Asf2 is uniformly continuous, for any a>0 there is b>0 such that p1−p2<b and p1,p2∈H1 imply that f2p1−f2p2<a. So, if(5)f1∘f2mh1−f1∘f2mh2<b,then(6)f2∘f1mf2h1−f2∘f1mf2h2<a.
Consequently, by(7)m∈ℕ:f1∘f2mh1−f1∘f2mh2<b∈G1,m∈ℕ:f2∘f1mf2h1−f2∘f1mf2h2<a∈G1.
By the definition, for anyh1,h2∈H1 with h1≠h2 there is δ>0 such that(8)m∈ℕ:f1∘f2mh1−f1∘f2mh2>δ∈G2.
Asf1 is uniformly continuous, for the above δ>0 there is δ′>0 such that p1−p2≤δ′ and p1,p2∈H2 imply that f1p1−f1p2≤δ. So, if(9)f1∘f2mh1−f1∘f2mh2>δ,then(10)f2∘f1m−1f2h1−f2∘f1m−1f2h2>δ′.
AsG2 is translation-invariant, by(11)m∈ℕ:f1∘f2mh1−f1∘f2mh2>δ∈G2,m−1∈ℕ:f2∘f1m−1f2h1−f2∘f1m−1f2h2>δ′∈G2.
This means that(12)m∈ℕ:f2∘f1mf2h1−f2∘f1mf2h2>δ′∈G2.
Thus, Theorem 1 is true.Theorem 2.
Let the product metricξ on the product space H1×H2 be defined by ξa1,b1,a2,b2=maxa2−a1,b2−b1 and the product map π1×π2 of π1:H1⟶H1 and π2:H2⟶H2 be defined by π1×π2a,b=π1a,π2b for any a∈H1 and any b∈H2, where H1,H2⊂ℝ are compact intervals, and let ϒa,b=f1b,f2a be a Cournot map. If G1 and G2 are two Furstenberg families such that G2 is translation-invariant, then f1∘f2 is strong G1,G2-chaotic if and only if so is f2∘f1.Proof.
Suppose thatf1∘f2 is strong G1,G2-chaotic. By the definition, there is an uncountable strong G1,G2-scrambled set D⊂H1 of f1∘f2. By the definition, for any given b>0 and any h1,h2∈H1 with h1≠h2 one has that(13)m∈ℕ:f1∘f2mh1−f1∘f2mh2<b∈G1.
Asf2 is uniformly continuous, for any a>0 there is b>0 such that p1−p2<b and p1,p2∈H1 imply that f2p1−f2p2<a. So, if(14)f1∘f2mh1−f1∘f2mh2<b,then(15)f2∘f1mf2h1−f2∘f1mf2h2<a.
Consequently, by(16)m∈ℕ:f1∘f2mh1−f1∘f2mh2<b∈G1,m∈ℕ:f2∘f1mf2h1−f2∘f1mf2h2<a∈G1.
By the definition, for anyh1,h2∈H1 with h1≠h2 there is δ>0 satisfying that for any h1,h2∈H1 with h1≠h2, one has that(17)m∈ℕ:f1∘f2mh1−f1∘f2mh2>δ∈G2.
Asf1 is uniformly continuous, for the above δ>0 there is δ′>0 such that p1−p2≤δ′ and p1,p2∈H2 imply that f1p1−f1p2≤δ. So, if(18)f1∘f2mh1−f1∘f2mh2>δ,then(19)f2∘f1m−1f2h1−f2∘f1m−1f2h2>δ′.
AsG2 is translation-invariant, by(20)m∈ℕ:f1∘f2mh1−f1∘f2mh2>δ∈G2,m−1∈ℕ:f2∘f1m−1f2h1−f2∘f1m−1f2h2>δ′∈G2.
This means that(21)m∈ℕ:f2∘f1mf2h1−f2∘f1mf2h2>δ′∈G2.
Thus, Theorem 2 is true.Corollary 1.
Letϒa,b=f1b,f2a be a Cournot map on the product space H1×H2. Then, for any a,b∈0,1, f1∘f2 is M¯a,M¯b-chaotic (resp. strong M¯a,M¯b-chaotic) if and only if so is f2∘f1.Proof.
AsM¯t is a translation-invariant Furstenberg family for any t∈0,1, by Theorems 1 and 2 one can see that Corollary 1 holds.Theorem 3.
Letϒa,b=f1b,f2a be a Cournot map on the product space H1×H2. If G1 and G2 are two Furstenberg families such that G2 is translation-invariant and satisfy that for any k∈1,2 and any G∈Gk, there is j∈0,1 satisfying that G2,j≔i∈0,1,…:2j+i∈G∈Gk, and that for any k∈1,2 and any G∈Gk,(22)G2≔2i+j∈0,1,…:j∈0,1,i∈G∈Gk,then ϒ is G1,G2-chaotic if and only if so is ϒ2Λ2.Proof.
We assume thatϒ is G1,G2-chaotic.Claim 1.
ϒ
2 is G1,G2-chaotic.The Proof of Claim 1.
Assume thatD⊂H1×H2 is a G1,G2-scrambled set of the system H1×H2,ϒ. As ϒ and ϒ2 are uniformly continuous, for any t>0 there is t′>0 satisfying that h1,h2∈H1×H2 and ξh1,h2<t′ imply ξϒh1,ϒh2<t and ξϒ2h1,ϒ2h2<t. By the hypothesis and the definition, for any d1,d2∈D with d1≠d2, one has that(23)G=m∈0,1,…:ξϒmd1,ϒmd2<t′∈G1.
AsG1 satisfies that for any G∈G1, there is j∈0,1 satisfying that G2,j≔i∈0,1,…:2j+i∈G∈G1, by the definition there is j∈0,1 satisfying that(24)G2,j≔i∈0,1,…:2j+i∈G∈G1.
By the above argument, one has that(25)G2,j⊂m∈0,1,…:ξϒ2m+j+2−jd1,ϒ2m+j+2−jd2<t.
That is,(26)G2,j⊂m∈0,1,…:ξϒ2m+2d1,ϒ2m+2d2<t.
So,(27)m∈0,1,…:ξϒ2m+2d1,ϒ2m+2d2<t∈G1.
AsG1 is translation-invariant,(28)m∈0,1,…:ξϒ2md1,ϒ2md2<t∈G1.
By the hypothesis and the definition, for any givend1,d2∈D with d1≠d2 there is δ>0 satisfying that(29)G′=m∈0,1,…:ξϒmd1,ϒmd2>δ∈G2.
AsG2 satisfies that for any G∈G2, there is j∈0,1 satisfying that G2,j≔i∈0,1,…:2j+i∈G∈G2, by the definition there is j∈0,1 satisfying that(30)G2,j′≔i∈0,1,…:2j+i∈G′∈G2.
Asϒ and ϒ2 are uniformly continuous, for the above δ>0, there is δ′>0 satisfying that h1,h2∈H1×H2 and ξh1,h2≤δ′ imply ξϒh1,ϒh2≤δ and ξϒ2h1,ϒ2h2≤δ. Clearly,(31)G2,j′⊂m∈0,1,…:ξϒ2md1,ϒ2md2>δ′,which means that(32)m∈0,1,…:ξϒ2md1,ϒ2md2>δ′∈G2.
Thus, Claim 1 holds.
Asϒ2=f1∘f2×f2∘f1, by hypothesis, Claim 1, the definition of G1,G2-chaos, and Theorem 1 and its proof, one can easily verify that f1∘f2 and f2∘f1 are G1,G2-chaotic.
Assume thatf1∘f2 is G1,G2-chaotic. By the definition, there is an uncountable subset C⊂H1 which is G1,G2-scrambled set of f1∘f2. By the proof of Theorem 1, f2C is an uncountable G1,G2-scrambled set of f2∘f1. Set A=a,g2a:a∈C. Then, A is uncountable. By the above argument, the definition of G1,G2-chaos and the proof of Theorem 1, it is easy to prove that A is a G1,G2-scrambled set of ϒ2Λ2.
Now, we assume thatϒ2Λ2 is G1,G2-chaotic.Claim 2.
ϒ is G1,G2-chaotic.The Proof of Claim 2.
By the hypothesis and the definitions,ϒ2 is G1,G2-chaotic. Assume that D is a G1,G2-scrambled set of the system H1×H2,ϒ2. As ϒn is uniformly continuous for any n∈0,1, for any t>0 there is t′>0 satisfying that h1,h2∈H1×H2 and ξh1,h2<t′ imply ξϒnh1,ϒnh2<t for any n∈0,1. By hypothesis and the definition, for any d1,d2∈D with d1≠d2, one has that(33)G=m∈0,1,…:ξϒ2md1,ϒ2md2<t′∈G1.
So, for anym∈G and any n∈0,1 we have that(34)ξϒ2m+nd1,ϒ2m+nd2<t.
AsG1 satisfies that for any G∈G1,(35)G2≔2i+j∈0,1,…:j∈0,1,i∈G∈G1,by the definition we have(36)G2≔2m+n:m∈G,n∈0,1∈G1.
Clearly,(37)G2⊂m∈0,1,…:ξϒmd1,ϒmd2<t.
This means that(38)m∈0,1,…:ξϒmd1,ϒmd2<t∈G1.
By the hypothesis and the definition, for any givend1,d2∈D with d1≠d2 there is δ′>0 satisfying that(39)G′=m∈0,1,…:ξϒmd1,ϒmd2>δ′∈G2.
Asϒn is uniformly continuous for any n∈0,1, for the above δ′>0 there is δ>0 satisfying that h1,h2∈H1×H2 and ξϒ2h1,ϒ2h2>δ imply ξϒnh1,ϒnh2>δ for any n∈0,1. So, for any m∈G′ and any n∈0,1 we have that(40)ξϒ2m−1+nd1,ϒ2m−1+nd2>δ.
AsG2 is translation-invariant, G′−1∈G2. As G2 satisfies that for any G∈G2,(41)G2≔2i+j∈0,1,…:j∈0,1,i∈G∈G2,G′−12≔2m−1+n:m−1∈G′−1,n∈0,1∈G2.
Clearly,(42)G′−12⊂m∈0,1,…:ξϒmd1,ϒmd2>δ,which means that(43)m∈0,1,…:ξϒmd1,ϒmd2>δ∈G2.
Thus, Claim 2 holds.
Consequently, Theorem 3 is true.Theorem 4.
Letϒa,b=f1b,f2a be a Cournot map on the product space H1×H2. If G1 and G2 are two Furstenberg families such that G2 is translation-invariant and satisfy that for any k∈1,2 and any G∈Gk, there is j∈0,1 satisfying that G2,j≔i∈0,1,…:2j+i∈G∈Gk, and that for any k∈1,2 and any G∈Gk,(44)G2≔2i+j∈0,1,…:j∈0,1,i∈G∈Gk,then ϒ is strong G1,G2-chaotic if and only if so is ϒ2Λ2.Proof.
We assume thatϒ is strong G1,G2-chaotic.Claim 3.
ϒ
2 is strong G1,G2-chaotic.The Proof of Claim 3.
Assume thatD⊂H1×H2 is a strong G1,G2-scrambled set of the system H1×H2,ϒ. As ϒ and ϒ2 are uniformly continuous, for any t>0 there is t′>0 satisfying that h1,h2∈H1×H2 and ξh1,h2<t′ imply ξϒh1,ϒh2<t and ξϒ2h1,ϒ2h2<t. By hypothesis and the definition, for any d1,d2∈D with d1≠d2, one has that(45)G=m∈0,1,…:ξϒmd1,ϒmd2<t′∈G1.
AsG1 satisfies that for any G∈G1,(46)G2≔2i+j∈0,1,…:j∈0,1,i∈G∈G1,by the definition there is j∈0,1 satisfying that(47)G2,j≔i∈0,1,…:2j+i∈G∈G1.
By the above argument, one has that(48)G2,j⊂m∈0,1,…:ξϒ2m+j+2−jd1,ϒ2m+j+2−jd2<t.
That is,(49)G2,j⊂m∈0,1,…:ξϒ2m+2d1,ϒ2m+2d2<t.
So,(50)m∈0,1,…:ξϒ2m+2d1,ϒ2m+2d2<t∈G1.
AsG1 is translation-invariant,(51)m∈0,1,…:ξϒ2md1,ϒ2md2<t∈G1.
By the hypothesis and the definition, there isδ>0 such that for any d1,d2∈D with d1≠d2,(52)G′=m∈0,1,…:ξϒmd1,ϒmd2>δ∈G2.
AsG2 satisfies that for any G∈G2, there is j∈0,1 satisfying that G2,j≔i∈0,1,…:2j+i∈G∈G2, by the definition there is j∈0,1 satisfying that(53)G2,j′≔i∈0,1,…:2j+i∈G′∈G2.
Asϒ and ϒ2 are uniformly continuous, for the above δ>0 there is δ′>0 satisfying that h1,h2∈H1×H2 and ξh1,h2≤δ′ imply ξϒh1,ϒh2≤δ and ξϒ2h1,ϒ2h2≤δ. Clearly,(54)G2,j′⊂m∈0,1,…:ξϒ2md1,ϒ2md2>δ′,which means that(55)m∈0,1,…:ξϒ2md1,ϒ2md2>δ′∈G2.
Thus, Claim 3 holds.
Asϒ2=f1∘f2×f2∘f1, by hypothesis, Claim 3, the definition of strong G1,G2-chaos, and Theorem 2 and its proof, one can easily verify that f1∘f2 and f2∘f1 are strong G1,G2-chaotic.
Assume thatf1∘f2 is strong G1,G2-chaotic. By the definition, there is an uncountable subset C⊂H1 which is strong G1,G2-scrambled set of f1∘f2. By the proof of Theorem 2, f2C is an uncountable and strong G1,G2-scrambled set of f2∘f1. Set A=a,g2a:a∈C. Then, A is uncountable. By the above argument, the definition of strong G1,G2-chaos and the proof of Theorem 2, it is easy to prove that A is a strong G1,G2-scrambled set of ϒ2Λ2.
Now, we assume thatϒ2Λ2 is strong G1,G2-chaotic.Claim 4.
ϒ is strong G1,G2-chaotic.The Proof of Claim 4.
By the hypothesis and the definitions,ϒ2 is strong G1,G2-chaotic. Assume that D is a strong G1,G2-scrambled set of the system H1×H2,ϒ2. As ϒn is uniformly continuous for any n∈0,1, for any t>0 there is t′>0 satisfying that h1,h2∈H1×H2 and ξh1,h2<t′ imply ξϒnh1,ϒnh2<t for any n∈0,1. By the hypothesis and the definition, for any d1,d2∈D with d1≠d2, one has that(56)G=m∈0,1,…:ξϒ2md1,ϒ2md2<t′∈G1.
So, for anym∈G and any n∈0,1, we have that(57)ξϒ2m+nd1,ϒ2m+nd2<t.
AsG1 satisfies that for any G∈G1,(58)G2≔2i+j∈0,1,…:j∈0,1,i∈G∈G1,by the definition we have(59)G2≔2m+n:m∈G,n∈0,1∈G1.
Clearly,(60)G2⊂m∈0,1,…:ξϒmd1,ϒmd2<t.
This means that(61)m∈0,1,…:ξϒmd1,ϒmd2<t∈G1.
By the hypothesis and the definition, there isδ′>0 satisfying that for any d1,d2∈D with d1≠d2,(62)G′=m∈0,1,…:ξϒmd1,ϒmd2>δ′∈G2.
Asϒn is uniformly continuous for any n∈0,1, for the above δ′>0 there is δ>0 satisfying that h1,h2∈H1×H2 and ξϒ2h1,ϒ2h2>δ imply ξϒnh1,ϒnh2>δ for any n∈0,1. So, for any m∈G′ and any n∈0,1, we have that(63)ξϒ2m−1+nd1,ϒ2m−1+nd2>δ.
AsG2 is translation-invariant, G′−1∈G2. As G2 satisfies that for any G∈G2,(64)G2≔2i+j∈0,1,…:j∈0,1,i∈G∈G2,G′−12≔2m−1+n:m−1∈G′−1,n∈0,1∈G2.
Clearly,(65)G′−12⊂m∈0,1,…:ξϒmd1,ϒmd2>δ,which means that(66)m∈0,1,…:ξϒmd1,ϒmd2>δ∈G2.
Thus, Claim 4 holds.
Consequently, Theorem 4 is true.Corollary 2.
Letϒa,b=f1b,f2a be a Cournot map on the product space H1×H2. Then, for any a,b∈0,1, ϒ is M¯a,M¯b-chaotic (resp. strong M¯a,M¯b-chaotic) if and only if so is ϒ2Λ2.Proof.
We have the following two claims.Claim 5.
For anyt∈0,1, M¯t satisfies that for any G∈G, there is j∈0,1 such that G2,j≔i∈0,1,…:2j+i∈G∈G.The Proof of Claim 5.
It is clear thatM¯0=ℬ and that if G∈ℬ, then there is j∈0,1 satisfying that G2,j∈ℬ. Assume that there is t∈0,1 such that M¯t does not have the property P2. By this assumption and the definition, there is G∈M¯t such that for any j∈0,1, μ¯G2,j=ej<t. Choose δj∈0,t−ej for any j∈0,1. As μ¯G2,j=ej<t for any j∈0,1, by the definition there is an integer M>0 such that for any j∈0,1 and any integer m≥M, ♯G2,j∩1,2,…,m<t−δjm. This implies that(67)♯1,2,…\G2,j∩1,2,…,m>m−t−δjm.
Letn∈2M+1,2M+1,⋯ and write n=2n/2+ln, where n/2 is the integral part of n/2 and ln∈0,1. By the definition, i∉G2,j implies 2i+j∉G for any j∈0,1. Obviously, if j1,j2∈0,1 and j1≠j2 then 2i1+j1≠2i2+j2 for any i1,i2∈0,1,…,m. So,(68)♯1,2,…\G∩1,2,…,n≥♯A0+♯A1>n2−t−e0n2+n2−t−e1n2,where(69)A0=1,2,…\G2,0∩1,2,…,n2,A1=1,2,…\G2,1∩1,2,…,n2.
This implies that(70)♯G∩1,2,…,n<n−n2−t−e0n2+n2−t−e1n2.
Setδ=minδ0,δ1. As(71)μ¯G=lim supn⟶∞1n♯G∩1,2,…,n≤lim supn⟶∞1nn−n2−t−e0n2+n2−t−e1n2,μ¯G≤1nn−2n2−t−δn2.
Byn=2n/2+ln,(72)μ¯G≤t−δ.
This is a contraction. Consequently, Claim 5 holds.Claim 6.
For anyt∈0,1, M¯t satisfies that for any G∈G:(73)G2≔2i+j∈0,1,…:j∈0,1,i∈G∈G.The Proof of Claim 6.
It is obvious thatM¯0=ℬ, and that if G∈ℬ, then G2∈ℬ. Assume that there is t∈0,1 such that M¯t does not have the property Q2. By this assumption and the definition, there is G∈M¯t such that μ¯G2=e<t. Choose δ∈0,t−e. As μ¯G2=e<t, by the definition there is an integer M>0 such that for any integer m≥M, ♯G2∩1,2,…,m<t−δm. Let n∈2M+1,2M+1,⋯ and write n=2n/2+ln, where n/2 is the integral part of n/2 and ln∈0,1. By the definition, i∈G implies 2i+j∈G2 for any j∈0,1. Obviously, if i1,i2∈G and j1≠j2, then 2i1+j1≠2i2+j2 for any j1,j2∈0,1. So,(74)2♯G∩1,2,…,n2≤♯G2∩1,2,…,n<nt−δ.
This implies that(75)♯G∩1,2,…,n2<122n2+lnt−δ.
Consequently,(76)μ¯G≤limn⟶∞12n/22n2+lnt−δ=t−δ<t.
This is a contraction. Consequently, Claim 5 is true.
From the above two claims we know thatM¯t satisfies the conditions of Theorems 3 and 4 for any t∈0,1. Thus, by these two theorems one can see that Corollary 2 holds.Theorem 5.
Letϒa,b=f1b,f2a be a Cournot map on the product space H1×H2. If G1 and G2 are two Furstenberg families such that G2 is translation-invariant, then ϒ is G1,G2-chaotic (resp. strong G1,G2-chaotic) if and only if so is ϒ2Λ1.Proof.
The proof is similar to those of Theorems 3 and 4 and is omitted.Corollary 3.
Letϒa,b=f1b,f2a be a Cournot map on the product space H1×H2. Then, for any a,b∈0,1, ϒ is M¯a,M¯b-chaotic (resp. strong M¯a,M¯b-chaotic) if and only if so is ϒ2Λ1.Proof.
By Theorem 5 and the proof of Corollary 2 one can easily see that Corollary 3 holds.Theorem 6.
Letϒa,b=f1b,f2a be a Cournot map on the product space H1×H2. If G1 and G2 are two Furstenberg families such that G2 is translation-invariant, then ϒ is G1,G2-chaotic (resp. strong G1,G2-chaotic) if and only if so is ϒ2Λ1∪Λ2.Proof.
By hypothesis, the definitions ofΛ1, Λ2, and G1,G2-chaos (resp. strong G1,G2-chaos) and Theorems 3 and 4, it is easily seen that ϒ is G1,G2-chaotic (resp. strong G1,G2-chaotic) if and only if so is ϒ2Λ1∪Λ2.Corollary 4.
Letϒa,b=f1b,f2a be a Cournot map on the product space H1×H2. Then, for any a,b∈0,1, ϒ is M¯a,M¯b-chaotic (resp. strong M¯a,M¯b-chaotic) if and only if so is ϒ2Λ1∪Λ2.Proof.
By Theorem 6 and the proof of Corollary 2 one can easily see that Corollary 4 holds.



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