# The Topological Entropy of Cyclic Permutation Maps and Some Chaotic Properties on Their MPE sets

Authors: Risong Li; Tianxiu Lu
Journal: Complexity
 (2020)

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## Abstract

In this paper, we study some chaotic properties ofs-dimensional dynamical system of the form Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1, where ak∈Hk for any k∈1,2,…,s,s≥2 is an integer, and Hk is a compact subinterval of the real line ℝ=−∞,+∞ for any k∈1,2,…,s. Particularly, a necessary and sufficient condition for a cyclic permutation map Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1 to be LY-chaotic or h-chaotic or RT-chaotic or D-chaotic is obtained. Moreover, the LY-chaoticity, h-chaoticity, RT-chaoticity, and D-chaoticity of such a cyclic permutation map is explored. Also, we proved that the topological entropy hΨ of such a cyclic permutation map is the same as the topological entropy of each of the following maps: gj∘gj−1∘⋯∘g1l∘gs∘gs−1∘⋯∘gj+1, if j=1,…,s−1and gs∘gs−1∘⋯∘g1, and that Ψ is sensitive if and only if at least one of the coordinates maps of Ψs is sensitive.


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## Body

## 1. Introduction

Letf:Y⟶X and g:X⟶Y be continuous maps on the compact subintervals X and Y of the real line ℝ, and let Φ:X×Y⟶X×Y be a continuous map defined by Φx,y=fy,gx for any x,y∈X×Y. These maps have been proposed to give a mathematical description of competition in a duopolistic market, called Cournot duopoly (see [1]). This is the reason why the above map Φ is called a Cournot map, and the above two maps f and g are called reaction functions (that is, the maps f and g give laws to organize the production of some firms which are competitors in a market).From [1, 2], we know that there are the so-called Markov perfect equilibria (MPE henceforth) processes, where the two players move alternatively such that each of them chooses the best reply to the previous action of another player. This occurs if the phase point xt,yt belongs alternatively to the graphs of the reaction curves y=gx and x=fy. This condition can be satisfied if the initial condition is a point of a reaction curve. That is, y0=gx0 (player 1 moves first) or x0=fy0 (player 2 moves first). This follows from the fact that the union of the graphs of the two reaction function is trapping for Φ.Probably, the first paper which gives the concept of chaos in a mathematically rigorous way is that of Li and Yorke [3]. Since then many different rigorous notions of chaos have been proposed. Each of these concepts tries to describe some kind of unpredictability in the evolution of the system. The notion of Li–Yorke sensitivity (LY-sensitivity) was presented for the first time by Akin and Kolyada in [4]. Moreover, they introduced the notion of spatiotemporal chaos. A very important generalization is distributional chaos, proposed by Schweizer and Smítal [5], mainly because it is equivalent to positive topological entropy and some other concepts of chaos when restricted to some spaces (see [5, 6]). It is noted that this equivalence does not transfer to higher dimensions, e.g., positive topological entropy does not imply distributional chaos in the case of triangular maps of the unit square [7] (the same happens when the dimension is zero [8]). In [9], Wang et al. introduced the concept of distributional chaos in a sequence and showed that it is equivalent to Li–Yorke chaos (LY-chaos) for continuous maps of the interval. During the last years, many researchers paid attention to the chaotic behavior of Cournot maps (see [1, 2, 10–17]).From [18], we know that if X is an infinite metric space, then if a continuous map f is transitive and has dense periodic points then it has sensitive dependence on initial conditions, which means that the third assumption in definition of chaos in sense of Devaney (D-chaos) is not necessary. By definition, it is clear that if a continuous map f is D-chaotic, then it is chaotic in sense of Ruelle–Takens (RT-chaotic). It was proved that the converse is false (see [13]). It is well known that D-chaotic maps are LY-chaotic (see [19]) and that maps with positive topological entropy (h-chaotic) are also LY-chaotic maps (see [20]). However, there exist LY-chaotic interval maps with zero topological entropy (see [21]). And from Theorem 1.2 in [13], we know that there exist LY-chaotic maps which are not D-chaotic. For interval continuous maps, J. S. Canovas and M. Ruiz Marin had the particular cases established in Theorem 1.2 which is from [13].LetW1=fy,y:y∈Y and W2=x,gx:x∈X The set W12=W1∪W2 represents the union of the graphs of the two reaction function and is trapping for Φ, i.e., ΦW12⊆W12. Canovas and Ruiz Marín called the set W12 a MPE set for Φ (see [13]). Moreover, they discussed and studied several chaotic properties (e.g., Devaney chaos, RT chaos, topological chaos, and Li–Yorke chaos) of Cournot maps and showed that it is not true that any chaotic property they considered satisfies the condition that Φ is chaotic if and only if ΦW12 is chaotic. Recently, Lu and Zhu further investigated the dynamical properties of Cournot maps, and more precisely, for the maps ΦW12, Φ2W1, and Φ2W2(see [15]). In [22], Linero Bas and Soler Lopez defined a cyclically permuted direct product map and studied the topics of (totally) topological transitivity and (weakly) topological mixing for cyclically permuted direct product maps by exploring the relationship between the dynamics of F and that of the compositions fσi∘⋯∘fσni, where i∈1,…,n, Fx1,x2,…,xn=fσ1xσ1,fσ2xσ2,…,fσnxσn (is called to be cyclically permuted direct product maps), which is defined from the Cartesian product X1×X2×…×Xn into itself, where X1,X2,…,Xn are general topological spaces, each map fσii:Xσi⟶Xi is continuous, i∈1,2,…,n, and σ is a cyclic permutation of 1,2,…,n,n>1. In [23], Linero Bas and Soler Lopez established some results on transitivity for cyclically permuted direct product maps of the Cartesian product In, where I=0,1. Particularly, it was shown that, for n≥3, the transitivity of this map F is equivalent to the total transitivity, and if n=2, they obtained a splitting result for transitive maps. Also, they extended well-known properties of transitivity from interval maps to cyclically permuted direct product maps. To do it, they used the strong link between the map F and the compositions ϕj=fσj∘⋯∘fσnj,j=1,…,n. For cyclically permuted direct product maps, if n=2 and X1=X2=0,1, these maps appears associated with certain economical model so called Cournot duopoly (see [2, 11–13], etc.). Even one can find them in age-structured population models, as in [24], where it is analyzed the Leslie model. For study on population models, we refer the reader to [24] and the references therein.For chaotic maps, there have been many applications. Since Li and Yorke [3] introduced the term of chaos in 1975, chaotic dynamical systems were highly discussed and investigated in the literature (see [18, 25, 26] and the references therein) as they are very good examples of problems coming from the theory of topological dynamics and model and many phenomena from biology, physics, chemistry, engineering, and social sciences. Recently, some new 1D and 2D chaotic systems with complex chaos performance have been developed (see [27–35] and the references therein).Motivated by [13, 15, 22, 23], we will deal with the dynamical properties of cyclic permutation maps:(1)Ψ:ΠHj⟶ΠHj,ψx1,…,xs=gsxs,g1x1,g2x2,…,gs−1xs−1,defined from a Cartesian product ΠHj=H1×⋯×Hs into itself, where each Hi is a compact subinterval of the real line, and gj:Hj⟶Hj+1,j=1,…,s−1.gs:HS⟶H1. Clearly, cyclic permutation map is a cyclically permuted direct product map. Since ΨS is a direct product of interval maps, Ψs=ϕ1×ϕ2×⋯×ϕs, with ϕj=gj−1∘⋯∘g1∘gs∘⋯∘gj for any j=2,⋯,s and ϕ1=gs∘gs−1∘⋯∘g1, it makes sense to analyze the dynamics of Ψ in terms of these compositions ϕj. In particular, a necessary and sufficient condition for a cyclic permutation map Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1 to be LY-chaotic or h-chaotic or RT-chaotic or D-chaotic is given. Moreover, the LY-chaoticity, h-chaoticity, RT-chaoticity, and D-chaoticity of such a cyclic permutation map is discussed. More precisely, for a cyclic permutation map,(2)Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1,where(3)a1,a2,…,as∈H1×H2×⋯×HS.We obtain the following results:(1)
Ψ is LY-chaotic if and only if φj is LY-chaotic for any j=1,…,s(2)
Ψ is h-chaotic if and only if φj is h-chaotic for any j=1,…,s(3)
IfΨ is RT-chaotic, then φj is RT-chaotic for any j=1,…,s(4)
IfΨ is D-chaotic, then φj is D-chaotic for any j=1,…,s(5)
Ψ is topologically transitive if and only if Ψ is D-chaotic if and only if Ψ is RT-chaotic(6)
IfΨ is topologically transitive, then it is h-chaotic(7)
IfΨ is D-chaotic, then ΨW12,…,s is D-chaotic(8)
IfΨ is RT-chaotic, then so is ΨW12,…,s(9)
ΨW12,…,s is h-chaotic if and only if so is Ψ(10)
ΨW12,…,s is LY-chaotic if and only if so is ΨOur results extend some existing ones on two-dimensional dynamical systems. Also, it is shown that the topological entropyhΨ of such a cyclic permutation map is the same as the topological entropy of each of the following maps: φj,j=1,…,s, and that it is sensitive if and only if at least one of the following maps is sensitive: φj,j=1,…,s.The interest of studying continuous cyclic permutation maps is as follows. Firstly, they ares-dimensional maps whose dynamical behavior is close to that of one-dimensional maps (see [22, 23, 36, 37] and the references therein), where s≥2 is any given integer, and their study could give us information about the dynamics of general s-dimensional maps. Secondly, these maps are closely related with the model of an economic process called Cournot duopoly (see [22–24, 38]), the age-structured population models (see [24] and the references therein), and the cyclically permuted direct product maps (see [22, 23] and the references therein). Thirdly, the chaotic dynamics is widely used in nonlinear control, synchronization communication, and many other applications (see [27–35] and the references therein).


## 2. Preliminaries

LetX be a metric space with metric d.A dynamical systemX,f or a continuous map f:X⟶X is said to be(1)
Transitive if for every pair of nonempty open setsU and V of X, there exists a positive integer n such that fnU∩V≠φ.(2)
Mixing if for every pair of nonempty open setsU and V of X, there exists a positive integer N such that fnU∩V≠φ for every integer n≥N.(3)
Sensitive if there is anε>0 such that whenever U is a nonempty open set of X, there exist points x,y∈U such that dfnx,fny>ε for some positive integer n, where ε is called a sensitive constant of f.(4)
Chaotic in the sense of Ruelle–Takens (or RT-chaotic, for short) (see [39]) if it is both transitive and sensitive.(5)
Chaotic in the sense of Li–Yorke (or LY-chaotic, for short) if there is an uncountable setD⊂X/Perf such that for any a,b∈D with a≠b:(4)liminfk⟶+∞dfka,fkb=0,limsupk⟶+∞dfka,fkb>0,wherePerf is the set of periodic points of f and a point x∈X is called a periodic point of f if there is an integer k>0 such that fkx=x.(6)
Chaotic in the sense of Devaney (or D-chaotic, for short) iff is transitive and sensitive with Perf¯=X, A¯ denotes the closure of the set A.A subsetD⊂X is said to be an n,ε-separated if for any a,b∈D with a≠b there is i∈0,1,…,n−1 with dfia,fib>ε. Let snε,f be the maximal possible cardinality of an n,ε-separated set. The topological entropy of f (see [25, 40]) is defined as(5)hf=limε⟶0limsupn⟶+∞1nlogsnε,f.Recall thathfn=nhf for any integer n≥0. A dynamical system X,f or a map f:X⟶X is h-chaotic if hf>0.LetHk be a compact interval of the real line ℝ=−∞,+∞ for any k∈1,2,…,s and H1×H2×⋯×HS endowed with the product metric σ which is defined by(6)σa1,a2,…,as,b1,b2,…,bs=max1≤k≤sbk−ak,for any(7)a1,a2,…,as,b1,b2,…,bs∈H1×H2×⋯×Hs.For any continuous mapf:X⟶X on a metric space X and any x∈X, the set of limit points of the sequence fmxm≥0∞ is the ω-limit set of x which is written by ωfx. Then, if X has no isolated points, then the map f is transitive if there exists x∈X with ωfx=X. Write Trf=x∈X:ωfx=X. Then, Trf is the set of transitive points of f.


## 3. Main Results

### 3.1. Relation between Some Chaotic Properties of a Continuous Cyclic Permutation Map and the Corresponding Properties of Every Coordinate Map

It is known in the frame of general topological spaces (and for general cyclic permutations) that ifΨ is transitive (even mixing or weakly mixing or totally transitive) then the associate compositions φjj=1,2,…,s so are. This was done in [22]. For completeness, we give the proof of Lemma 1.We need the following two lemmas.Lemma 1.
For a continuous cyclic permutation map,(8)Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1,where(9)a1,a2,…,as∈H1×H2×⋯×HS.
If it is topologically transitive, then every coordinate map ofΨs is topologically transitive.Proof.
As the proof of the result in Lemma1 is similar for any s≥3, for simplicity we only prove Lemma 1 for s=3. Suppose that Ψ is topologically transitive and let(10)a1,a2,a3∈H1×H2×.H3,with(11)ωΨa1,a2,a3=H1×H2×.H3,where ωΨa1,a2,a3 is the ω-limit set of a1,a2,a3 under Ψ. By (3) in [36], hypothesis, the definition,(12)Ψ3nx,y,z=g3∘g2∘g1x,g1∘g3∘g2y,g2∘g1∘g3z,Ψ3n+1=g3∘g2∘g1∘g3z,g1∘g3∘g2∘g1x,g2∘g1∘g3∘g2y,Ψ3n+2=g3∘g2∘g1∘g3∘g2y,g1∘g3∘g2∘g1∘g3z,g2∘g1∘g3∘g2∘g1x.
One can easily prove that(13)H1×H2×.H3=ωΨa1,a2,…,as,⊂ωg3∘g2∘g1a1×ωg1∘g3∘g2a2×ωg2∘g1∘g3a3,∪ωg3∘g2∘g1g3a3×ωg1∘g3∘g2g1a1×ωg2∘g1∘g3g2a2,∪ωg3∘g2∘g1g3g2a2×ωg1∘g3∘g2g1g3a3×ωg2∘g1∘g3g2g1a1,⊂H1×H2×.H3.
Now, we show that if(14)ωg3∘g2∘g1a1=H1,then(15)ωg1∘g3∘g2g1a1=H2,ωg2∘g1∘g3g2g1a1=H3.
AsΨ is topologically transitive, gi is surjective for any i∈1,2,3. So, for any y2∈H2 there is y1∈H1 with g1y1=y2. Let mii=1∞ be a sequence of positive integers with(16)limi⟶∞g3∘g2∘g1mia1=y1.
By the continuity ofg1,(17)limi⟶∞g1g3∘g2∘g1mia1=g1y1,That is, limi⟶∞g3∘g2∘g1mig1a1=y2. This implies that(18)ωg1∘g3∘g2g1a1=H2.
Similarly, one can prove(19)ωg2∘g1∘g3g2g1a1=H3.
By the similar argument, we obtain that if(20)ωg1∘g3∘g2a2=H2,then(21)ωg2∘g1∘g3g2a2=H3,ωg3∘g2∘g1g3g2a2=H1,and that if(22)ωg2∘g1∘g3a3=H3,then(23)ωg3∘g2∘g1g3a3=H1,ωg1∘g3∘g2g1g3a3=H2.
By the above argument and the transitivity ofΨ, it follows that the case ωg3∘g2∘g1a1≠H1,ωg1∘g3∘g2a2≠H2, and ωg2∘g1∘g3a3≠H3 cannot happen. Thus, by the above argument, we have(24)ωg3∘g2∘g1a1=H1,ωg1∘g3∘g2a2=H2,ωg2∘g1∘g3a3=H3.
This means thatg3∘g2∘g1, g1∘g3∘g2, and g2∘g1∘g3 are topologically transitive.
However, it is not true that the transitivity ofϕj implies the transitivity of Ψ. In fact, we can construct examples of maps Ψ which are not transitive but the compositions ϕj are transitive. Let us show an example.Example 1.
LetΨ=y,z,fx:In⟶In,n=3, with f:I⟶I transitive but not totally transitive, that is, f2I is not transitive (here, I=0,1 is the unit interval, but we can translate the example to arbitrary sets H1×H2×H3). Sincef2I is not transitive, we can decompose the unit interval into I=I1∪I2, being I1andI2 compact intervals such that I1∩I2=u, with fu=u and such thatfI1=I2,fI2=I1, andf2I mixing, j=1,2(for details, consult, for instance, [26]). In this case, Ψ6x,y,z=f2x,f2y,f2z, and taking into account that f2Ij=Ij, we find that Ψ6I1×I1×I1⊆I1×I1×I1⊆I3. Consequently, Ψ6 is not transitive. Due to the fact that n≥3, according to [23] (see Corollary B), we have that Ψ is transitive if and only if Ψ is totally transitive; since Ψ6 is not transitive, we deduce that Ψ is not transitive although its associated composition ϕj=f is. Nevertheless, in the two-dimensional case, when Ψx,y=fy,gx, and H1=H2=I, the transitivity of f∘g implies the transitivity of Ψ, as it was proved in Lemma 2.1 in [13] (Canovas-Marin).
We note that the following lemma holds in the general setting of topological spaces: to this end, consult Lemma 7 in [23]. So, the proof of Lemma 2 is omitted here.Lemma 2.
For a topologically transitive cyclic permutation map,(25)Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1,where(26)a1,a2,…,as∈H1×H2×⋯×HS,Perϕj¯=Hj for any j=1,…,s. Then, one has(27)PerΨ¯=H1×H2×⋯×HS.Theorem 1.
For a cyclic permutation map,(28)Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1,where(29)a1,a2,…,as∈H1×H2×⋯×HS.The following hold:(1)
Ψ is LY-chaotic if and only if φj is LY-chaotic for any j=1,…,s(2)
Ψ is h-chaotic if and only if φj is h-chaotic for any j=1,…,s(3)
IfΨ is RT-chaotic, then φj is RT-chaotic for any j=1,…,s(4)
IfΨ is D-chaotic, then φj is D-chaotic for any j=1,…,s(5)
Ifφj is RT-chaotic for any j=1,…,s and satisfies that there is an transitive point of hj∈Hj for any j=1,…,s such that(30)ωΨh1,h2,…,hs=ωφ1h1×ωφ2h2×…×ωφshs,∪ωφ1gshs×ωϕ2g1h1×⋯×ωφsgs−1hs−1,∪ωφ1gs∘gs−1hs−1×ωφ2g1∘gshs×⋯×ωφsgs−1∘gs−2hs−2,∪ωφ1gs∘gs−1∘gs−2hs−2×ωφ2g1∘gs∘gs−1hs−1×⋯×ωφsgs−1∘gs−2∘gs−3hs−3,∪ωφ1gs∘gs−1∘⋯∘g3h3×ωφ2g1∘gs∘gs−1∘⋯∘g4h4×⋯×ωφs−2gs−3∘gs−4∘⋯∘g2∘g1∘gshs×ωφs−1gs−2∘⋯∘g1h1×ωφsgs−1∘gs−2∘⋯∘g2h2,∪ωφ1gs∘gs−1∘g2h2×ωφ2g1∘gs∘⋯∘g3h3×⋯×ωφs−1gs−2∘g1∘gshs×⋯×ωφsgs−1∘gs−2∘g1h1.thenΨ is RT-chaotic.(6)
Ifφj is D-chaotic for any j=1,…,s and satisfies that there is an transitive point of hj∈Hj for any j=1,…,s such that(31)ωΨh1,h2,…,hs=ωφ1h1×ωφ2h2×⋯×ωφshs,∪ωφ1gshs×ωφ2g1h1×⋯×ωφsgs−1hs−1,∪ωφ1gs∘gs−1hs−1×ωφ2g1∘gshs×⋯×ωφsgs−1∘gs−2hs−2,∪ωφ1gs∘gs−1∘gs−2hs−2×ωφ2g1∘gs∘gs−1hs−1×⋯×ωφsgs−1∘gs−2∘gs−3hs−3,∪ωφ1gs∘gs−1∘⋯∘g3h3×ωφ2g1∘gs∘gs−1∘⋯∘g4h4×⋯×ωφs−2gs−3∘gs−4∘⋯∘g2∘g1∘gshs×ωφs−1gs−2∘⋯∘g1h1×ωφsgs−1∘gs−2∘⋯∘g2h2,∪ωφ1gs∘gs−1∘g2h2×ωφ2g1∘gs∘⋯∘g3h3×⋯×ωφs−1gs−2∘g1∘gshs×⋯×ωφsgs−1∘gs−2∘g1h1.Then,Ψ is D-chaotic.(7)
Ifϕj is RT-chaotic for any j=1,…,s and satisfies that, for any j=1,…,s and any hj∈Hj,(32)ωΨh1,h2,…,hs⊂ωφ1h1×ωφ2h2×⋯×ωφshs,∪ωφ1gshs×ωφ2g1h1×⋯×ωφsgs−1hs−1,∪ωφ1gs∘gs−1hs−1×ωφ2g1∘gshs×⋯×ωφsgs−1∘gs−2hs−2,∪ωφ1gs∘gs−1∘gs−2hs−2×ωφ2g1∘gs∘gs−1hs−1×⋯×ωφsgs−1∘gs−2∘gs−3hs−3,∪⋯,∪ωφ1gs∘gs−1∘⋯∘g3h3×ωφ2g1∘gs∘gs−1∘⋯∘g4h4×⋯×ωφs−2gs−3∘gs−4∘⋯∘g2∘g1∘gshs×ωφs−1gs−2∘⋯∘g1h1×ωφsgs−1∘gs−2∘⋯∘g2h2,∪ωφ1gs∘gs−1∘g2h2×ωφ2g1∘gs∘⋯∘g3h3×⋯×ωφs−1gs−2∘g1∘gshs×⋯×ωφsgs−1∘gs−2∘g1h1,whereA⊂B means that A⊆B and A≠B; then, Ψ is not RT-chaotic.(8)
Ifφj is D-chaotic for any j=1,…,s and satisfies that, for any j=1,…,s and any hj∈Hj,(33)ωΨh1,h2,…,hs⊂ωφ1h1×ωφ2h2×⋯×ωφshs,∪ωφ1gshs×ωφ2g1h1×⋯×ωφsgs−1hs−1,∪ωφ1gs∘gs−1hs−1×ωφ2g1∘gshs×⋯×ωφsgs−1∘gs−2hs−2,∪ωφ1gs∘gs−1∘gs−2hs−2×ωφ2g1∘gs∘gs−1hs−1×⋯×ωφsgs−1∘gs−2∘gs−3hs−3,∪⋯,∪ωφ1gs∘gs−1∘⋯∘g3h3×ωφ2g1∘gs∘gs−1∘⋯∘g4h4×⋯×ωφs−2gs−3∘gs−4∘⋯∘g2∘g1∘gshs×ωφs−1gs−2∘⋯∘g1h1×ωφsgs−1∘gs−2∘⋯∘g2h2,∪ωφ1gs∘gs−1∘g2h2×ωφ2g1∘gs∘⋯∘g3h3×⋯×ωφs−1gs−2∘g1∘gshs×⋯×ωφsgs−1∘gs−2∘g1h1,where A⊂B means that A⊆B and A≠B, then Ψ is not D-chaotic.Proof.
By Propositions 3.1 and 3.2 in [15], the definition, hypothesis, and(34)Ψs=φ1×φ2×⋯×φs.
One can easily verify that statement (1) is true. By [41], the definition, and Ψs=φ1×φ2×⋯×φs, we can easily prove that (2) is true. Now, we show that statement (4) is true. Suppose that Ψ is D-chaotic, that is, it is topologically transitive and sensitive such that(35)PerΨ¯=PerΨs¯=H1×H2×⋯×HS.
As(36)Ψs=φ1×φ2×⋯×φs,and PerΨs=PerΨ, by Lemma 1, Theorem 1.2 in [13, 42] if Ψ is D-chaotic then so is ϕj for any j=1,…,s.
Next, we show that statement (3) is also true. Suppose thatΨ is RT-chaotic. Then, it is transitive and sensitive. By Lemmas 1 and 2, one has that ϕj is topologically transitive for any j=1,…,s such that Perϕj¯=Hj for any j=1,…,s. By Theorem 1.2 in [13], ϕj is sensitive for any j=1,…,s, which implies that ϕj is RT-chaotic for any j=1,…,s.
Finally, by the proof of Lemma 2.1 in [13], Lemma 1 in [43], the definitions, and hypothesis, statements (5), (6), (7), and (8) are true.Theorem 2.
For a cyclic permutation map,(37)Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1,where(38)a1,a2,…,as∈H1×H2×⋯×HS,the following holds:(1)
Ψ is topologically transitive if and only if Ψ is D-chaotic if and only if Ψ is RT-chaotic(2)
IfΨ is topologically transitive then it is h-chaoticProof 3.
(1)
IfΨ is D-chaotic, then by definition it is RT-chaotic. If Ψ is RT-chaotic, then it is topologically transitive. Therefore, it is enough to prove that if Ψ is topologically transitive, then it is D-chaotic. Now, we let Ψ be topologically transitive. By Lemma 2,(39)PerΨ¯=PerΨs¯=H1×H2×⋯×HS,which means thatΨ is sensitive. So, Ψ is D-chaotic.(2)
Suppose thatΨ is topologically transitive. By Lemma 1, ϕj is topologically transitive for any j=1,…,s. As(40)Ψs=φ1×φ2×⋯×φs,by Theorem 1.2 in [13] hΨs>0. This implies that hΨ>0.


### 3.2. Chaos on MPE Set for a Permutation Map

For a permutation map,(41)Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1,where(42)a1,a2,…,as∈H1×H2×⋯×HS.Let(43)Ws=gsas,g1∘gsas,g2∘g1∘gsas,…,gs−2∘⋯∘g1∘gsas,as:as∈Hs,Ws−1=gsbs,g1∘gsbs,g2∘g1∘gsbs,…,gs−3∘⋯∘g1∘gsbs,as−1,bs:as−1∈Hs−1,⋮W1=a1,g1a1,…,gs−1∘⋯∘g1a1:a1∈H1,where bs=gs−1as−1. Let W12,…,s=W1∪W2∪…∪WS be the MPE set for Ψ.Theorem 3.
For a permutation map,(44)Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1,where(45)a1,a2,…,as∈H1×H2×⋯×HS,the following hold:(1)
IfΨ is D-chaotic, then ΨW12,…,s is D-chaotic(2)
IfΨ is RT-chaotic, then so is ΨW12,…,s(3)
ΨW12,…,s is h-chaotic if and only if so is Ψ(4)
ΨW12,…,s is LY-chaotic if and only if so is ΨProof 4.
As the proof of the result in Theorem3 is similar for any s≥3, for simplicity, we only prove Theorem 3 for s=3. First, we claim that if Ψ is topologically transitive then so is ΨW123. To prove this, we let a1,a2,a3∈H1×H2×H3 with ωΨa1,a2,a3∈H1×H2×H3. By the proof of Lemma 1, one can easily see that a1∈Trg3∘g2∘g1 or a2∈Trg1∘g3∘g2 or a3∈Trg2∘g1∘g3. By the proof of Lemma 1, if a1∈Trg3∘g2∘g1, then g1a1∈Trg1∘g3∘g2 and g2∘g1a1∈Trg2∘g1∘g3. Fix a∈H1. Then, a,g1a,g2∘g1a∈W1. By hypothesis and the definition, there is a sequence mii=1∞ of positive integers with(46)limi⟶∞Ψmia,g1a,g2∘g1a=a,g1a,g2∘g1a.
Fixb∈H2. Then, g3∘g2b,b,g2b∈W2. As g1 is onto, there is a∈H1 with g1a=b. As g1a1∈Trg1∘g3∘g2,(47)limi⟶∞Ψmi+1a1,g1a1,g2∘g1a1,=limi⟶∞Ψmig3∘g2∘g1a1,g1a1,g2∘g1a1,=g3∘g2b,b,gsb.
Fixc∈H3. Then, g3c,g1∘g3c,c∈W3. As g1 and g2 is onto, there is a∈H1 with g2∘g1a=c. As g∘2g1a1∈Trg2∘g1∘g3,(48)limi⟶∞Ψmi+2a1,g1a1,g2∘g1a1,=limi⟶∞Ψmig3∘g2∘g1a1,g1∘g3∘g2∘g1a1,g2∘g1a1,=g3c,g1∘g3c,c.(1)
Suppose thatΨ is D-chaotic. By the above argument, ΨW123 is topologically transitive. As ΨW123⊆W123, by hypothesis and the definition ΨW123 is D-chaotic.(2)
Suppose thatΨ is RT-chaotic, which implies that it is topologically transitive. By the above argument, ΨW123 is topologically transitive. Let a∈H1. Then, a,g1a,g2∘g1a∈W1. By Theorem 1, there exists δ>0 such that, for any open neighborhood V of a, there exist b∈V and an integer m≥0 with(49)g3∘g2∘g1ma−g3∘g2∘g1mb>δ.This means that(50)σΨ3ma,g1a,g2∘g1a,Ψ3mb,g1b,g2∘g1b>δ.Similarly, the condition can be proved for anyg3∘g2a,a,g2a∈W2 and any g3a,g1∘g3a,a∈W3. By the definition, ΨW123 is RT-chaotic.(3)
Clearly,hΨW123>0 implies that hΨ>0. Suppose that hΨ>0. Consider Ψ3W123. It is easy to see that W1 is trapping for Ψ3 and that(51)hΨ3W1=hg3∘g2∘g1+hg1∘g3∘g2g1H1+hg2∘g1∘g3g2∘g1H1.As(52)hΨ=hg3∘g2∘g1=hg1∘g3∘g2g1H1=hg2∘g1∘g3|g2∘g1H1>0,hΨ3W123>0, which implies that hΨW123>0.(4)
Obviously, by the definition, ifΨW123 is LY-chaotic then so is Ψ. Suppose that Ψ is LY-chaotic. By Theorem 1, g3∘g2∘g1,g1∘g3∘g2, and g2∘g1∘g3 are LY-chaotic. Let D be an uncountable Li–Yorke chaotic set for g3∘g2∘g1 and let(53)B=a,g1a,g2∘g1a:a∈D.
Then,B⊂W1 is uncountable. Now, we will prove that B is a Li–Yorke chaotic set for Ψ3W1. For any(54)a,g1a,g2∘g1a,b,g1b,g2∘g1b∈B,with a≠b, one has that(55)limsupm⟶∞σΨ3ma,g1a,g2∘g1a,Ψ3mb,g1b,g2∘g1b,≥limsupm⟶∞g3∘g2∘g1ma−g3∘g2∘g1mb>0.
Since(56)liminfm⟶∞g3∘g2∘g1ma−g3∘g2∘g1mb=0,there is an increasing sequence mii=1∞ of positive integers with(57)limi⟶∞g3∘g2∘g1mig1a−g3∘g2∘g1mig1b=0.
Asg1 and g2 are uniformly continuous,(58)limi⟶∞g3∘g2∘g1mig1a−g3∘g2∘g1migb1=0,(59)limi⟶∞|g3∘g2∘g1mig2∘g1a−g3∘g2∘g1mig2∘gb1|=0,which mean that(60)liminfm⟶∞σΨ3ma,g1a,g2∘g1a,Ψ3mb,g1b,g2∘g1b,≤liminfi⟶∞σΨ3mia,g1a,g2∘g1a,Ψ3mib,g1b,g2∘g1b,≤limi⟶∞g3∘g2∘g1mia−g3∘g2∘g1mib+g3∘g2∘g1mig1a−g3∘g2∘g1mig1b+g3∘g2∘g1mig∘2g1a−g3∘g2∘g1mig∘2g1b=0.
Consequently,B is a Li–Yorke chaotic set for Ψ3W1, which implies that Ψ3W1 is LY-chaotic. Thus, ΨW123 is LY-chaotic.


### 3.3. The Topological Entropy and Sensitivity of a Continuous Cyclic Permutation Map

For the propertieshΨn=nhΨ and hF×G=hF+hG, we refer the reader to [41]. And for the property(61)hgj∘gj−1∘⋯∘g1∘gs∘gs−1∘⋯∘gj+1=hgs∘gs−1∘⋯∘g2∘g1,we refer the reader to [44], where it is proved, in the general setting of continuous selfmaps f1,…,fm of a compact topological space X that(62)hfm∘⋯∘f2∘f1=hfj−1∘⋯∘f2∘f1∘fm∘⋯∘fj,j=2,…,m.Moreover, the topological entropy of this kind of maps has been already computed in [45]. However, for completeness, we give Theorem 4 and its proof here.Theorem 4.
For a continuous cyclic permutation map,(63)Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1,where(64)a1,a2,…,as∈H1×H2×⋯×HS,hΨ=hϕj for any j=1,…,s.Proof.
As(65)Ψs=φ1×φ2×⋯×φs.by [38] and Proposition 5.2 in [2], Theorem 4 holds.
For the sensitive properties of product maps or semiflows, we refer the reader to [43, 46]. However, for completeness, we give Theorem.5 and its proof here.Theorem 5.
For a continuous cyclic permutation map,(66)Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1,where(67)a1,a2,…,as∈H1×H2×⋯×HS,and it is sensitive if and only if at least one of the following maps is sensitive: ϕj,j=1,…,s.Proof.
As(68)Ψs=φ1×φ2×⋯×φs,by Lemma 1 in [43] and Theorem 1 in [47], Theorem 5 holds.


### 3.4. Discussion on Applications

It is clear that a continuous cyclic permutation map defined by(69)Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1,for any(70)a1,a2,…,as∈H1×H2×⋯×HS,in this paper is a cyclically permuted direct product (see [22, 23] and references therein). For cyclically permuted direct products, there have been many important results and applications (see [22, 23] and references therein).Whens=2 and H1=H2=0,1, this type of continuous cyclic permutation map appears as certain economical model so called Cournot duopoly (see [1, 2, 10–17, 38, 42] and references therein). For many important results and applications of Cournot duopoly games, we refer the reader to [1, 2, 10–17, 38, 42] and the references therein.From [31], we know that one can find continuous cyclic permutation maps in age-structured population models, as in [24], where it is analyzed as the Leslie model:(71)y1n+1=yNgyN,y2n+1=y1n,…,yNn+1=yN−1n,where g is a C1-map and each variable yin,i=1,…,N,n=0,1,2,…, determines the population size of the j-age class in the nth period, being yi0 the initial population. To study some dynamical behavior of this kind of model is equivalent to explore the same behavior of the c.p.d.p. map Fy1,y2,…,yN=yNgyN,y1,…,yN−1 (which is also a continuous cyclic permutation map), where σi=i−1modN,i=1,…,N. For study, results and applications of population models, we refer the reader to [24] and the references therein.Because the chaotic maps have the excellent properties of unpredictability, ergodicity, and sensitivity to their parameters and initial values, they are widely used in security applications. In [35], Zhou et al. introduced a simple and effective chaotic system by a combination of two existing one-dimension (1D) chaotic maps which are called seed maps. By simulations and performance evaluations, it was shown that the proposed system can produce lots of 1D chaotic maps having larger chaotic ranges and better chaotic behaviors compared with their seed maps. To explore its applications in multimedia security, a novel image encryption algorithm is presented. By using a same set of security keys, this algorithm can generate a completely different encrypted image each time when it is used to the same original image. By experiments and security analysis it was shown that the algorithm has excellent performance in image encryption and various attacks. In [27], Hua et al. introduced a new two-dimensional Sine Logistic modulation map (2D-SLMM) which is given by the Logistic and Sine maps. When it is compared with existing chaotic maps, we can find that it has the better chaotic range, better ergodicity, hyperchaotic property, and relatively low implementation cost. Also, to investigate its applications, they proposed a chaotic magic transform (CMT) to efficiently change the image pixel positions. Combining 2D-SLMM with CMT, they further introduced a new image encryption algorithm. Simulation results and security analysis show that this algorithm can present images with low time complexity and a high security level as well as to resist various attacks. In [32], Wu et al. Noonan introduced a number of Sudoku-associated matrix element representations besides the conventional representation by matrix row-column pair. Particularly, they are representations via Sudoku matrix row-digit pair, digit-row pair, column-digit pair, digit-column pair, block-digit pair, and digit-block pair. So, one can secretly represent matrix elements by a Sudoku matrix and develop new Sudoku associated 2D parametric bijections. To show the effectiveness and randomness of thes bijections, they introduced a simple and effective Sudoku Associated Image Scrambler by 2D Sudoku-associated bijections for image scrambling without bandwidth expansion. By simulations and comparisons, it was demonstrated that the proposed method can outperform some state-of-the-art methods. In [28], Hua and Zhou gave a two-dimensional Logistic-adjusted-Sine map (2D-LASM). By performance evaluations, it was shown that this map has better ergodicity and unpredictability, and a wider chaotic range than many the existing chaotic maps. By this map, they further designed a 2D-LASM-based image encryption scheme (LAS-IES). The principle of diffusion and confusion are strictly finished, and a mechanism of adding random values to plain image is established to enhance the security level of cipher image. By simulation results and security analysis, it was shown that the LAS-IES can efficiently encrypt different kinds of images into random-like ones which have strong ability of resisting various security attacks. In [29], Hua et al. gave a sine-transform-based chaotic system (STBCS) which generates one-dimensional (1-D) chaotic maps. This chaotic system performs a sine transform to the combination of the outputs of the two existing chaotic maps (seed maps). Users have the flexibility to pick any existing 1D chaotic maps as seed maps in STBCS to generate lots of new chaotic maps. The complex chaotic behavior of STBCS is verified by using the principle of Lypunov exponent. To show the usability and effectiveness of STBCS, they presented three new chaotic maps as examples. By theoretical analysis, it was shown that these chaotic maps have complex dynamics properties and robust chaos. Performance evaluations show that they have better chaotic ranges, better complexity, and unpredictability, compared with chaotic maps generated by other methods and the corresponding seed maps. Also, to explain the simplicity of STBCS in hardware implementation, they simulated the three new chaotic maps by the field-programmable gate array (FPGA). In [30], Hua et al. presented a two-dimensional (2D) sine chaotification system (2D-SCS). Such a system can not only significantly enhance the complexity of 2D chaotic maps but also greatly extend their chaotic ranges. As examples, they applied 2D-SCS to two existing 2D chaotic maps to get two enhanced chaotic maps. By performance evaluations, it was shown that these two enhanced chaotic maps have robust chaotic behaviors in much larger chaotic ranges than the existing 2D chaotic maps. Also, a microcontroller-based experiment platform was designed to implement these enhanced chaotic maps in hardware devices. Furthermore, to discuss the application of 2D-SCS, these two enhanced chaotic maps are used to design a pseudorandom number generator. By experiment results, one can see that these enhanced chaotic maps are able to produce better random sequences than the existing 2D and several state-of-the-art one-dimensional (1D) chaotic maps. In [31], Hua et al. Zhou gave a two-dimensional (2D) modular chaotification system (2D-MCS) to improve the chaos complexity of any 2D chaotic map. Since the modular operation is a bounded transform, the chaotic maps which are improved by 2D-MCS can generate chaotic behaviors in wide parameter ranges while the existing chaotic maps cannot. Three improved chaotic maps were given as typical examples to verify the effectiveness of 2D-MCS. The chaotic properties of one example of 2D-MCS are mathematically analyzed by using Lyapunov exponent. By performance evaluations, it was shown that these improved chaotic maps have continuous and large chaotic ranges, and their outputs are distributed more uniformly than the outputs of the existing 2D chaotic maps. To explain the application of 2D-MCS, they applied the improved chaotic maps of 2D-MCS to secure communication. By the simulation results, it was shown that these improved chaotic maps exhibit better performance than a few existing and newly developed chaotic maps in terms of resisting different channel noise.In the future, we will further discuss and explore some properties and applications of continuous cyclic permutation maps [48].


## 3.1. Relation between Some Chaotic Properties of a Continuous Cyclic Permutation Map and the Corresponding Properties of Every Coordinate Map

It is known in the frame of general topological spaces (and for general cyclic permutations) that ifΨ is transitive (even mixing or weakly mixing or totally transitive) then the associate compositions φjj=1,2,…,s so are. This was done in [22]. For completeness, we give the proof of Lemma 1.We need the following two lemmas.Lemma 1.
For a continuous cyclic permutation map,(8)Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1,where(9)a1,a2,…,as∈H1×H2×⋯×HS.
If it is topologically transitive, then every coordinate map ofΨs is topologically transitive.Proof.
As the proof of the result in Lemma1 is similar for any s≥3, for simplicity we only prove Lemma 1 for s=3. Suppose that Ψ is topologically transitive and let(10)a1,a2,a3∈H1×H2×.H3,with(11)ωΨa1,a2,a3=H1×H2×.H3,where ωΨa1,a2,a3 is the ω-limit set of a1,a2,a3 under Ψ. By (3) in [36], hypothesis, the definition,(12)Ψ3nx,y,z=g3∘g2∘g1x,g1∘g3∘g2y,g2∘g1∘g3z,Ψ3n+1=g3∘g2∘g1∘g3z,g1∘g3∘g2∘g1x,g2∘g1∘g3∘g2y,Ψ3n+2=g3∘g2∘g1∘g3∘g2y,g1∘g3∘g2∘g1∘g3z,g2∘g1∘g3∘g2∘g1x.
One can easily prove that(13)H1×H2×.H3=ωΨa1,a2,…,as,⊂ωg3∘g2∘g1a1×ωg1∘g3∘g2a2×ωg2∘g1∘g3a3,∪ωg3∘g2∘g1g3a3×ωg1∘g3∘g2g1a1×ωg2∘g1∘g3g2a2,∪ωg3∘g2∘g1g3g2a2×ωg1∘g3∘g2g1g3a3×ωg2∘g1∘g3g2g1a1,⊂H1×H2×.H3.
Now, we show that if(14)ωg3∘g2∘g1a1=H1,then(15)ωg1∘g3∘g2g1a1=H2,ωg2∘g1∘g3g2g1a1=H3.
AsΨ is topologically transitive, gi is surjective for any i∈1,2,3. So, for any y2∈H2 there is y1∈H1 with g1y1=y2. Let mii=1∞ be a sequence of positive integers with(16)limi⟶∞g3∘g2∘g1mia1=y1.
By the continuity ofg1,(17)limi⟶∞g1g3∘g2∘g1mia1=g1y1,That is, limi⟶∞g3∘g2∘g1mig1a1=y2. This implies that(18)ωg1∘g3∘g2g1a1=H2.
Similarly, one can prove(19)ωg2∘g1∘g3g2g1a1=H3.
By the similar argument, we obtain that if(20)ωg1∘g3∘g2a2=H2,then(21)ωg2∘g1∘g3g2a2=H3,ωg3∘g2∘g1g3g2a2=H1,and that if(22)ωg2∘g1∘g3a3=H3,then(23)ωg3∘g2∘g1g3a3=H1,ωg1∘g3∘g2g1g3a3=H2.
By the above argument and the transitivity ofΨ, it follows that the case ωg3∘g2∘g1a1≠H1,ωg1∘g3∘g2a2≠H2, and ωg2∘g1∘g3a3≠H3 cannot happen. Thus, by the above argument, we have(24)ωg3∘g2∘g1a1=H1,ωg1∘g3∘g2a2=H2,ωg2∘g1∘g3a3=H3.
This means thatg3∘g2∘g1, g1∘g3∘g2, and g2∘g1∘g3 are topologically transitive.
However, it is not true that the transitivity ofϕj implies the transitivity of Ψ. In fact, we can construct examples of maps Ψ which are not transitive but the compositions ϕj are transitive. Let us show an example.Example 1.
LetΨ=y,z,fx:In⟶In,n=3, with f:I⟶I transitive but not totally transitive, that is, f2I is not transitive (here, I=0,1 is the unit interval, but we can translate the example to arbitrary sets H1×H2×H3). Sincef2I is not transitive, we can decompose the unit interval into I=I1∪I2, being I1andI2 compact intervals such that I1∩I2=u, with fu=u and such thatfI1=I2,fI2=I1, andf2I mixing, j=1,2(for details, consult, for instance, [26]). In this case, Ψ6x,y,z=f2x,f2y,f2z, and taking into account that f2Ij=Ij, we find that Ψ6I1×I1×I1⊆I1×I1×I1⊆I3. Consequently, Ψ6 is not transitive. Due to the fact that n≥3, according to [23] (see Corollary B), we have that Ψ is transitive if and only if Ψ is totally transitive; since Ψ6 is not transitive, we deduce that Ψ is not transitive although its associated composition ϕj=f is. Nevertheless, in the two-dimensional case, when Ψx,y=fy,gx, and H1=H2=I, the transitivity of f∘g implies the transitivity of Ψ, as it was proved in Lemma 2.1 in [13] (Canovas-Marin).
We note that the following lemma holds in the general setting of topological spaces: to this end, consult Lemma 7 in [23]. So, the proof of Lemma 2 is omitted here.Lemma 2.
For a topologically transitive cyclic permutation map,(25)Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1,where(26)a1,a2,…,as∈H1×H2×⋯×HS,Perϕj¯=Hj for any j=1,…,s. Then, one has(27)PerΨ¯=H1×H2×⋯×HS.Theorem 1.
For a cyclic permutation map,(28)Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1,where(29)a1,a2,…,as∈H1×H2×⋯×HS.The following hold:(1)
Ψ is LY-chaotic if and only if φj is LY-chaotic for any j=1,…,s(2)
Ψ is h-chaotic if and only if φj is h-chaotic for any j=1,…,s(3)
IfΨ is RT-chaotic, then φj is RT-chaotic for any j=1,…,s(4)
IfΨ is D-chaotic, then φj is D-chaotic for any j=1,…,s(5)
Ifφj is RT-chaotic for any j=1,…,s and satisfies that there is an transitive point of hj∈Hj for any j=1,…,s such that(30)ωΨh1,h2,…,hs=ωφ1h1×ωφ2h2×…×ωφshs,∪ωφ1gshs×ωϕ2g1h1×⋯×ωφsgs−1hs−1,∪ωφ1gs∘gs−1hs−1×ωφ2g1∘gshs×⋯×ωφsgs−1∘gs−2hs−2,∪ωφ1gs∘gs−1∘gs−2hs−2×ωφ2g1∘gs∘gs−1hs−1×⋯×ωφsgs−1∘gs−2∘gs−3hs−3,∪ωφ1gs∘gs−1∘⋯∘g3h3×ωφ2g1∘gs∘gs−1∘⋯∘g4h4×⋯×ωφs−2gs−3∘gs−4∘⋯∘g2∘g1∘gshs×ωφs−1gs−2∘⋯∘g1h1×ωφsgs−1∘gs−2∘⋯∘g2h2,∪ωφ1gs∘gs−1∘g2h2×ωφ2g1∘gs∘⋯∘g3h3×⋯×ωφs−1gs−2∘g1∘gshs×⋯×ωφsgs−1∘gs−2∘g1h1.thenΨ is RT-chaotic.(6)
Ifφj is D-chaotic for any j=1,…,s and satisfies that there is an transitive point of hj∈Hj for any j=1,…,s such that(31)ωΨh1,h2,…,hs=ωφ1h1×ωφ2h2×⋯×ωφshs,∪ωφ1gshs×ωφ2g1h1×⋯×ωφsgs−1hs−1,∪ωφ1gs∘gs−1hs−1×ωφ2g1∘gshs×⋯×ωφsgs−1∘gs−2hs−2,∪ωφ1gs∘gs−1∘gs−2hs−2×ωφ2g1∘gs∘gs−1hs−1×⋯×ωφsgs−1∘gs−2∘gs−3hs−3,∪ωφ1gs∘gs−1∘⋯∘g3h3×ωφ2g1∘gs∘gs−1∘⋯∘g4h4×⋯×ωφs−2gs−3∘gs−4∘⋯∘g2∘g1∘gshs×ωφs−1gs−2∘⋯∘g1h1×ωφsgs−1∘gs−2∘⋯∘g2h2,∪ωφ1gs∘gs−1∘g2h2×ωφ2g1∘gs∘⋯∘g3h3×⋯×ωφs−1gs−2∘g1∘gshs×⋯×ωφsgs−1∘gs−2∘g1h1.Then,Ψ is D-chaotic.(7)
Ifϕj is RT-chaotic for any j=1,…,s and satisfies that, for any j=1,…,s and any hj∈Hj,(32)ωΨh1,h2,…,hs⊂ωφ1h1×ωφ2h2×⋯×ωφshs,∪ωφ1gshs×ωφ2g1h1×⋯×ωφsgs−1hs−1,∪ωφ1gs∘gs−1hs−1×ωφ2g1∘gshs×⋯×ωφsgs−1∘gs−2hs−2,∪ωφ1gs∘gs−1∘gs−2hs−2×ωφ2g1∘gs∘gs−1hs−1×⋯×ωφsgs−1∘gs−2∘gs−3hs−3,∪⋯,∪ωφ1gs∘gs−1∘⋯∘g3h3×ωφ2g1∘gs∘gs−1∘⋯∘g4h4×⋯×ωφs−2gs−3∘gs−4∘⋯∘g2∘g1∘gshs×ωφs−1gs−2∘⋯∘g1h1×ωφsgs−1∘gs−2∘⋯∘g2h2,∪ωφ1gs∘gs−1∘g2h2×ωφ2g1∘gs∘⋯∘g3h3×⋯×ωφs−1gs−2∘g1∘gshs×⋯×ωφsgs−1∘gs−2∘g1h1,whereA⊂B means that A⊆B and A≠B; then, Ψ is not RT-chaotic.(8)
Ifφj is D-chaotic for any j=1,…,s and satisfies that, for any j=1,…,s and any hj∈Hj,(33)ωΨh1,h2,…,hs⊂ωφ1h1×ωφ2h2×⋯×ωφshs,∪ωφ1gshs×ωφ2g1h1×⋯×ωφsgs−1hs−1,∪ωφ1gs∘gs−1hs−1×ωφ2g1∘gshs×⋯×ωφsgs−1∘gs−2hs−2,∪ωφ1gs∘gs−1∘gs−2hs−2×ωφ2g1∘gs∘gs−1hs−1×⋯×ωφsgs−1∘gs−2∘gs−3hs−3,∪⋯,∪ωφ1gs∘gs−1∘⋯∘g3h3×ωφ2g1∘gs∘gs−1∘⋯∘g4h4×⋯×ωφs−2gs−3∘gs−4∘⋯∘g2∘g1∘gshs×ωφs−1gs−2∘⋯∘g1h1×ωφsgs−1∘gs−2∘⋯∘g2h2,∪ωφ1gs∘gs−1∘g2h2×ωφ2g1∘gs∘⋯∘g3h3×⋯×ωφs−1gs−2∘g1∘gshs×⋯×ωφsgs−1∘gs−2∘g1h1,where A⊂B means that A⊆B and A≠B, then Ψ is not D-chaotic.Proof.
By Propositions 3.1 and 3.2 in [15], the definition, hypothesis, and(34)Ψs=φ1×φ2×⋯×φs.
One can easily verify that statement (1) is true. By [41], the definition, and Ψs=φ1×φ2×⋯×φs, we can easily prove that (2) is true. Now, we show that statement (4) is true. Suppose that Ψ is D-chaotic, that is, it is topologically transitive and sensitive such that(35)PerΨ¯=PerΨs¯=H1×H2×⋯×HS.
As(36)Ψs=φ1×φ2×⋯×φs,and PerΨs=PerΨ, by Lemma 1, Theorem 1.2 in [13, 42] if Ψ is D-chaotic then so is ϕj for any j=1,…,s.
Next, we show that statement (3) is also true. Suppose thatΨ is RT-chaotic. Then, it is transitive and sensitive. By Lemmas 1 and 2, one has that ϕj is topologically transitive for any j=1,…,s such that Perϕj¯=Hj for any j=1,…,s. By Theorem 1.2 in [13], ϕj is sensitive for any j=1,…,s, which implies that ϕj is RT-chaotic for any j=1,…,s.
Finally, by the proof of Lemma 2.1 in [13], Lemma 1 in [43], the definitions, and hypothesis, statements (5), (6), (7), and (8) are true.Theorem 2.
For a cyclic permutation map,(37)Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1,where(38)a1,a2,…,as∈H1×H2×⋯×HS,the following holds:(1)
Ψ is topologically transitive if and only if Ψ is D-chaotic if and only if Ψ is RT-chaotic(2)
IfΨ is topologically transitive then it is h-chaoticProof 3.
(1)
IfΨ is D-chaotic, then by definition it is RT-chaotic. If Ψ is RT-chaotic, then it is topologically transitive. Therefore, it is enough to prove that if Ψ is topologically transitive, then it is D-chaotic. Now, we let Ψ be topologically transitive. By Lemma 2,(39)PerΨ¯=PerΨs¯=H1×H2×⋯×HS,which means thatΨ is sensitive. So, Ψ is D-chaotic.(2)
Suppose thatΨ is topologically transitive. By Lemma 1, ϕj is topologically transitive for any j=1,…,s. As(40)Ψs=φ1×φ2×⋯×φs,by Theorem 1.2 in [13] hΨs>0. This implies that hΨ>0.


## 3.2. Chaos on MPE Set for a Permutation Map

For a permutation map,(41)Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1,where(42)a1,a2,…,as∈H1×H2×⋯×HS.Let(43)Ws=gsas,g1∘gsas,g2∘g1∘gsas,…,gs−2∘⋯∘g1∘gsas,as:as∈Hs,Ws−1=gsbs,g1∘gsbs,g2∘g1∘gsbs,…,gs−3∘⋯∘g1∘gsbs,as−1,bs:as−1∈Hs−1,⋮W1=a1,g1a1,…,gs−1∘⋯∘g1a1:a1∈H1,where bs=gs−1as−1. Let W12,…,s=W1∪W2∪…∪WS be the MPE set for Ψ.Theorem 3.
For a permutation map,(44)Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1,where(45)a1,a2,…,as∈H1×H2×⋯×HS,the following hold:(1)
IfΨ is D-chaotic, then ΨW12,…,s is D-chaotic(2)
IfΨ is RT-chaotic, then so is ΨW12,…,s(3)
ΨW12,…,s is h-chaotic if and only if so is Ψ(4)
ΨW12,…,s is LY-chaotic if and only if so is ΨProof 4.
As the proof of the result in Theorem3 is similar for any s≥3, for simplicity, we only prove Theorem 3 for s=3. First, we claim that if Ψ is topologically transitive then so is ΨW123. To prove this, we let a1,a2,a3∈H1×H2×H3 with ωΨa1,a2,a3∈H1×H2×H3. By the proof of Lemma 1, one can easily see that a1∈Trg3∘g2∘g1 or a2∈Trg1∘g3∘g2 or a3∈Trg2∘g1∘g3. By the proof of Lemma 1, if a1∈Trg3∘g2∘g1, then g1a1∈Trg1∘g3∘g2 and g2∘g1a1∈Trg2∘g1∘g3. Fix a∈H1. Then, a,g1a,g2∘g1a∈W1. By hypothesis and the definition, there is a sequence mii=1∞ of positive integers with(46)limi⟶∞Ψmia,g1a,g2∘g1a=a,g1a,g2∘g1a.
Fixb∈H2. Then, g3∘g2b,b,g2b∈W2. As g1 is onto, there is a∈H1 with g1a=b. As g1a1∈Trg1∘g3∘g2,(47)limi⟶∞Ψmi+1a1,g1a1,g2∘g1a1,=limi⟶∞Ψmig3∘g2∘g1a1,g1a1,g2∘g1a1,=g3∘g2b,b,gsb.
Fixc∈H3. Then, g3c,g1∘g3c,c∈W3. As g1 and g2 is onto, there is a∈H1 with g2∘g1a=c. As g∘2g1a1∈Trg2∘g1∘g3,(48)limi⟶∞Ψmi+2a1,g1a1,g2∘g1a1,=limi⟶∞Ψmig3∘g2∘g1a1,g1∘g3∘g2∘g1a1,g2∘g1a1,=g3c,g1∘g3c,c.(1)
Suppose thatΨ is D-chaotic. By the above argument, ΨW123 is topologically transitive. As ΨW123⊆W123, by hypothesis and the definition ΨW123 is D-chaotic.(2)
Suppose thatΨ is RT-chaotic, which implies that it is topologically transitive. By the above argument, ΨW123 is topologically transitive. Let a∈H1. Then, a,g1a,g2∘g1a∈W1. By Theorem 1, there exists δ>0 such that, for any open neighborhood V of a, there exist b∈V and an integer m≥0 with(49)g3∘g2∘g1ma−g3∘g2∘g1mb>δ.This means that(50)σΨ3ma,g1a,g2∘g1a,Ψ3mb,g1b,g2∘g1b>δ.Similarly, the condition can be proved for anyg3∘g2a,a,g2a∈W2 and any g3a,g1∘g3a,a∈W3. By the definition, ΨW123 is RT-chaotic.(3)
Clearly,hΨW123>0 implies that hΨ>0. Suppose that hΨ>0. Consider Ψ3W123. It is easy to see that W1 is trapping for Ψ3 and that(51)hΨ3W1=hg3∘g2∘g1+hg1∘g3∘g2g1H1+hg2∘g1∘g3g2∘g1H1.As(52)hΨ=hg3∘g2∘g1=hg1∘g3∘g2g1H1=hg2∘g1∘g3|g2∘g1H1>0,hΨ3W123>0, which implies that hΨW123>0.(4)
Obviously, by the definition, ifΨW123 is LY-chaotic then so is Ψ. Suppose that Ψ is LY-chaotic. By Theorem 1, g3∘g2∘g1,g1∘g3∘g2, and g2∘g1∘g3 are LY-chaotic. Let D be an uncountable Li–Yorke chaotic set for g3∘g2∘g1 and let(53)B=a,g1a,g2∘g1a:a∈D.
Then,B⊂W1 is uncountable. Now, we will prove that B is a Li–Yorke chaotic set for Ψ3W1. For any(54)a,g1a,g2∘g1a,b,g1b,g2∘g1b∈B,with a≠b, one has that(55)limsupm⟶∞σΨ3ma,g1a,g2∘g1a,Ψ3mb,g1b,g2∘g1b,≥limsupm⟶∞g3∘g2∘g1ma−g3∘g2∘g1mb>0.
Since(56)liminfm⟶∞g3∘g2∘g1ma−g3∘g2∘g1mb=0,there is an increasing sequence mii=1∞ of positive integers with(57)limi⟶∞g3∘g2∘g1mig1a−g3∘g2∘g1mig1b=0.
Asg1 and g2 are uniformly continuous,(58)limi⟶∞g3∘g2∘g1mig1a−g3∘g2∘g1migb1=0,(59)limi⟶∞|g3∘g2∘g1mig2∘g1a−g3∘g2∘g1mig2∘gb1|=0,which mean that(60)liminfm⟶∞σΨ3ma,g1a,g2∘g1a,Ψ3mb,g1b,g2∘g1b,≤liminfi⟶∞σΨ3mia,g1a,g2∘g1a,Ψ3mib,g1b,g2∘g1b,≤limi⟶∞g3∘g2∘g1mia−g3∘g2∘g1mib+g3∘g2∘g1mig1a−g3∘g2∘g1mig1b+g3∘g2∘g1mig∘2g1a−g3∘g2∘g1mig∘2g1b=0.
Consequently,B is a Li–Yorke chaotic set for Ψ3W1, which implies that Ψ3W1 is LY-chaotic. Thus, ΨW123 is LY-chaotic.


## 3.3. The Topological Entropy and Sensitivity of a Continuous Cyclic Permutation Map

For the propertieshΨn=nhΨ and hF×G=hF+hG, we refer the reader to [41]. And for the property(61)hgj∘gj−1∘⋯∘g1∘gs∘gs−1∘⋯∘gj+1=hgs∘gs−1∘⋯∘g2∘g1,we refer the reader to [44], where it is proved, in the general setting of continuous selfmaps f1,…,fm of a compact topological space X that(62)hfm∘⋯∘f2∘f1=hfj−1∘⋯∘f2∘f1∘fm∘⋯∘fj,j=2,…,m.Moreover, the topological entropy of this kind of maps has been already computed in [45]. However, for completeness, we give Theorem 4 and its proof here.Theorem 4.
For a continuous cyclic permutation map,(63)Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1,where(64)a1,a2,…,as∈H1×H2×⋯×HS,hΨ=hϕj for any j=1,…,s.Proof.
As(65)Ψs=φ1×φ2×⋯×φs.by [38] and Proposition 5.2 in [2], Theorem 4 holds.
For the sensitive properties of product maps or semiflows, we refer the reader to [43, 46]. However, for completeness, we give Theorem.5 and its proof here.Theorem 5.
For a continuous cyclic permutation map,(66)Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1,where(67)a1,a2,…,as∈H1×H2×⋯×HS,and it is sensitive if and only if at least one of the following maps is sensitive: ϕj,j=1,…,s.Proof.
As(68)Ψs=φ1×φ2×⋯×φs,by Lemma 1 in [43] and Theorem 1 in [47], Theorem 5 holds.


## 3.4. Discussion on Applications

It is clear that a continuous cyclic permutation map defined by(69)Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1,for any(70)a1,a2,…,as∈H1×H2×⋯×HS,in this paper is a cyclically permuted direct product (see [22, 23] and references therein). For cyclically permuted direct products, there have been many important results and applications (see [22, 23] and references therein).Whens=2 and H1=H2=0,1, this type of continuous cyclic permutation map appears as certain economical model so called Cournot duopoly (see [1, 2, 10–17, 38, 42] and references therein). For many important results and applications of Cournot duopoly games, we refer the reader to [1, 2, 10–17, 38, 42] and the references therein.From [31], we know that one can find continuous cyclic permutation maps in age-structured population models, as in [24], where it is analyzed as the Leslie model:(71)y1n+1=yNgyN,y2n+1=y1n,…,yNn+1=yN−1n,where g is a C1-map and each variable yin,i=1,…,N,n=0,1,2,…, determines the population size of the j-age class in the nth period, being yi0 the initial population. To study some dynamical behavior of this kind of model is equivalent to explore the same behavior of the c.p.d.p. map Fy1,y2,…,yN=yNgyN,y1,…,yN−1 (which is also a continuous cyclic permutation map), where σi=i−1modN,i=1,…,N. For study, results and applications of population models, we refer the reader to [24] and the references therein.Because the chaotic maps have the excellent properties of unpredictability, ergodicity, and sensitivity to their parameters and initial values, they are widely used in security applications. In [35], Zhou et al. introduced a simple and effective chaotic system by a combination of two existing one-dimension (1D) chaotic maps which are called seed maps. By simulations and performance evaluations, it was shown that the proposed system can produce lots of 1D chaotic maps having larger chaotic ranges and better chaotic behaviors compared with their seed maps. To explore its applications in multimedia security, a novel image encryption algorithm is presented. By using a same set of security keys, this algorithm can generate a completely different encrypted image each time when it is used to the same original image. By experiments and security analysis it was shown that the algorithm has excellent performance in image encryption and various attacks. In [27], Hua et al. introduced a new two-dimensional Sine Logistic modulation map (2D-SLMM) which is given by the Logistic and Sine maps. When it is compared with existing chaotic maps, we can find that it has the better chaotic range, better ergodicity, hyperchaotic property, and relatively low implementation cost. Also, to investigate its applications, they proposed a chaotic magic transform (CMT) to efficiently change the image pixel positions. Combining 2D-SLMM with CMT, they further introduced a new image encryption algorithm. Simulation results and security analysis show that this algorithm can present images with low time complexity and a high security level as well as to resist various attacks. In [32], Wu et al. Noonan introduced a number of Sudoku-associated matrix element representations besides the conventional representation by matrix row-column pair. Particularly, they are representations via Sudoku matrix row-digit pair, digit-row pair, column-digit pair, digit-column pair, block-digit pair, and digit-block pair. So, one can secretly represent matrix elements by a Sudoku matrix and develop new Sudoku associated 2D parametric bijections. To show the effectiveness and randomness of thes bijections, they introduced a simple and effective Sudoku Associated Image Scrambler by 2D Sudoku-associated bijections for image scrambling without bandwidth expansion. By simulations and comparisons, it was demonstrated that the proposed method can outperform some state-of-the-art methods. In [28], Hua and Zhou gave a two-dimensional Logistic-adjusted-Sine map (2D-LASM). By performance evaluations, it was shown that this map has better ergodicity and unpredictability, and a wider chaotic range than many the existing chaotic maps. By this map, they further designed a 2D-LASM-based image encryption scheme (LAS-IES). The principle of diffusion and confusion are strictly finished, and a mechanism of adding random values to plain image is established to enhance the security level of cipher image. By simulation results and security analysis, it was shown that the LAS-IES can efficiently encrypt different kinds of images into random-like ones which have strong ability of resisting various security attacks. In [29], Hua et al. gave a sine-transform-based chaotic system (STBCS) which generates one-dimensional (1-D) chaotic maps. This chaotic system performs a sine transform to the combination of the outputs of the two existing chaotic maps (seed maps). Users have the flexibility to pick any existing 1D chaotic maps as seed maps in STBCS to generate lots of new chaotic maps. The complex chaotic behavior of STBCS is verified by using the principle of Lypunov exponent. To show the usability and effectiveness of STBCS, they presented three new chaotic maps as examples. By theoretical analysis, it was shown that these chaotic maps have complex dynamics properties and robust chaos. Performance evaluations show that they have better chaotic ranges, better complexity, and unpredictability, compared with chaotic maps generated by other methods and the corresponding seed maps. Also, to explain the simplicity of STBCS in hardware implementation, they simulated the three new chaotic maps by the field-programmable gate array (FPGA). In [30], Hua et al. presented a two-dimensional (2D) sine chaotification system (2D-SCS). Such a system can not only significantly enhance the complexity of 2D chaotic maps but also greatly extend their chaotic ranges. As examples, they applied 2D-SCS to two existing 2D chaotic maps to get two enhanced chaotic maps. By performance evaluations, it was shown that these two enhanced chaotic maps have robust chaotic behaviors in much larger chaotic ranges than the existing 2D chaotic maps. Also, a microcontroller-based experiment platform was designed to implement these enhanced chaotic maps in hardware devices. Furthermore, to discuss the application of 2D-SCS, these two enhanced chaotic maps are used to design a pseudorandom number generator. By experiment results, one can see that these enhanced chaotic maps are able to produce better random sequences than the existing 2D and several state-of-the-art one-dimensional (1D) chaotic maps. In [31], Hua et al. Zhou gave a two-dimensional (2D) modular chaotification system (2D-MCS) to improve the chaos complexity of any 2D chaotic map. Since the modular operation is a bounded transform, the chaotic maps which are improved by 2D-MCS can generate chaotic behaviors in wide parameter ranges while the existing chaotic maps cannot. Three improved chaotic maps were given as typical examples to verify the effectiveness of 2D-MCS. The chaotic properties of one example of 2D-MCS are mathematically analyzed by using Lyapunov exponent. By performance evaluations, it was shown that these improved chaotic maps have continuous and large chaotic ranges, and their outputs are distributed more uniformly than the outputs of the existing 2D chaotic maps. To explain the application of 2D-MCS, they applied the improved chaotic maps of 2D-MCS to secure communication. By the simulation results, it was shown that these improved chaotic maps exhibit better performance than a few existing and newly developed chaotic maps in terms of resisting different channel noise.In the future, we will further discuss and explore some properties and applications of continuous cyclic permutation maps [48].



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