The aim of this paper is to prove the following results: a continuous
map f : [0; 1] ! [0; 1] is chaotic if the shift map of : lim
([0; 1]; f) ! lim
([0; 1]; f) is
chaotic. However, this result fails, in general, for arbitrary compact metric spaces.
 f : lim
([0; 1]; f) ! lim
([0; 1]; f) is chaotic i  there exists an increasing sequence
of positive integers A such that the topological sequence entropy hA( f ) > 0. Finally,
for any A there exists a chaotic continuous map fA : [0; 1] ! [0; 1] such that
hA( fA) = 0:This paper has been partially supported by the grant
PB/2/FS/97 (Fundación Séneca, Comunidad Autónoma de Murcia).
I wish to thank the referee for the proof of Proposition 2.1 and the comments
that helped me to improve the paper

Cánovas Peña, José Salvador

Repositorio Digital de la Universidad Politécnica de Cartagena

Acta Math. Univ. ComenianaeVol. LXVIII, 2(1999), pp. 205–211205ON TOPOLOGICAL SEQUENCE ENTROPY ANDCHAOTIC MAPS ON INVERSE LIMIT SPACESJ. S. CANOVASAbstract. The aim of this paper is to prove the following results: a continuousmap f : [0, 1]→ [0, 1] is chaotic iff the shift map σf : lim← ([0, 1], f)→ lim← ([0, 1], f) ischaotic. However, this result fails, in general, for arbitrary compact metric spaces.σf : lim← ([0, 1], f) → lim← ([0, 1], f) is chaotic iff there exists an increasing sequenceof positive integers A such that the topological sequence entropy hA(σf ) > 0. Fi-nally, for any A there exists a chaotic continuous map fA : [0, 1] → [0, 1] such thathA(σfA ) = 0.1. IntroductionLet (X, d) and f : X → X be a compact metric space and a continuous maprespectively. Consider the space of sequenceslim← (X, f) = {x = (x0, x1, . . . , xn, . . . ) : xi ∈ X, f(xi) = xi−1, for i = 1, 2, . . . }.This set is called the inverse limit space associated to X and f . Define a newmetric d˜ on lim← (X, f) asd˜(x, y) =∞∑i=0d(xi, yi)2i,where x = (x0, x1, . . . , xn, . . . ) and y = (y0, y1, . . . , yn, . . . ). Then (lim← (X, f), d˜)is a compact metric space. Consider the natural projection pi : lim← (X, f) → Xdefined by pi(x0, x1, . . . , xn, . . . ) = x0. Note that d˜(x, y) ≥ d(pi(x), pi(y)) for allx, y ∈ lim← (X, f). The shift map is a homeomorphism σf : lim← (X, f)→ lim← (X, f)defined byσf (x) = σf (x0, x1, . . . , xn, . . . ) = (f(x0), x0, x1, . . . , xn, . . . ).It is clear that pi ◦ σf = f ◦ pi.Received December 14, 1998; received March 25, 1999.1980 Mathematics Subject Classification (1991 Revision). Primary 58F03, 26A18.206 J. S. CANOVASInverse limit spaces have been studied in the setting of dynamical systems in alarge number of papers. In [6], Shihai Li proved that some dynamical propertieshold at the same time for f and σf . In particular, he showed that f is chaoticin Devaney’s sense iff σf is also like that. He also proved a similar result for asuitable definition of w-chaos. In this paper, a similar result is studied in case ofthe Li-Yorke’s chaos. Recall briefly this definition of chaos.A point p ∈ X is periodic if there exists a positive integer n such thatfn(p) = p. The smallest positive integer satisfying this condition is called theperiod of p. Denote by Per(f) the set of periodic points of f . A pointx ∈ X is said to be asymptotically periodic if there exists a p ∈ Per(f) suchthat lim supn→∞ d(fn(x), fn(p)) = 0. A map f : X → X is said to be chaoticin the sense of Li-Yorkeor simply chaotic if there exists an uncountable setD ⊂ X \ Per(f) such thatlim supn→∞d(fn(x), fn(y)) > 0,lim infn→∞ d(fn(x), fn(y)) = 0,hold for all x, y ∈ D, x 6= y. D is called a scrambled set of f .The Li-Yorke’s chaos on inverse limit spaces has been studied by Gu Rongbaoin [3]. In that paper the author attempts to prove that a continuous map f ischaotic iff the shift map σf is chaotic. However, in the proof he uses implicitlythat f is surjective. As we will see later, this hypothesis on f cannot be removedin the following theorem essentially proved in [3].Theorem 1.1. Suppose f that is surjective. Then it is chaotic in the sense ofLi-Yorke if and only if the map σf is chaotic in the sense of Li-Yorke.When continuous maps f : [0, 1] → [0, 1] are concerned, the Li-Yorke’s chaosis connected with the notion of topological sequence entropy. Let us recall thedefinition (see [2]). Let A = {ai}∞i=1 be an increasing sequence of positive inte-gers. Given  > 0, we say that E ⊂ X is an (A, , n, f)-separated set if for anyx, y ∈ E with x 6= y there exists 1 ≤ k ≤ n such that d(fak(x), fak(y)) > . De-note by sn(A, , f) the cardinality of any maximal (A, , n, f)-separated set. Thetopological sequence entropy of f is given byhA(f) = lim→0lim supn→∞1nlog sn(A, , f).In general,(1) hA(f) ≥ hA(σf )for every A. When f is surjective we obtain the equality (see [2])(2) hA(f) = hA(σf ).The connection between the Li-Yorke’s chaos and the topological sequence entropyis established in the following result (see [1] and [5]).ON SEQUENCE ENTROPY AND CHAOS 207Theorem 1.2. Let c, d ∈ R and let f : [c, d]→ [c, d] be continuous. Then(a) f is chaotic iff there exists an increasing sequence of positive integers Asuch that hA(f) > 0.(b) For any increasing sequence A there exists a chaotic map fA : [c, d]→ [c, d]such that hA(fA) = 0.Theorem 1.2(a) does not hold in general for continuous maps on arbitrary com-pact metric spaces as it can be seen in [8]. In that paper, on [0, 1]× [0, 1] a chaoticmap f with supA hA(f) = 0 and a non-chaotic map g with supA hA(g) > 0 areconstructed.The aim of this paper is to prove the following results: f : [0, 1]→ [0, 1] is chaoticiff σf is chaotic. Theorem 1.2 holds for maps σf : lim← ([0, 1], f) → lim← ([0, 1], f).Moreover, an example of a chaotic map f for which σf is not chaotic is given.2. Positive Results for One-Dimensional MapsLet f : [0, 1] → [0, 1] be continuous. Consider [a, b] = ⋂n≥0 fn[0, 1]. Thenf |[a,b] : [a, b]→ [a, b] is obviously surjective.Proposition 2.1. Under the above conditions f is chaotic iff f |[a,b] is chaotic.Proof. It is clear that if f |[a,b] is chaotic then f is chaotic. Suppose that fis chaotic and let D be a scrambled of f . It is easy to see that fn(D) is also ascrambled set of f . Let Dn = fn(D) ∩ [a, b]. If {fn(x) : n ≥ 0} ∩ [a, b] = ∅, thenx is asymptotically periodic and then x /∈ Dn for all n ∈ N. So, it must exist apositive integer n0 such that Dn0 is uncountable. Then, f |[a,b] is chaotic. Theorem 2.2. Let f : [0, 1]→ [0, 1] be continuous. Then:(a) f is chaotic if and only if σf is chaotic.(b) σf : lim← ([0, 1], f) → lim← ([0, 1], f) is chaotic if and only if there exists anincreasing sequence of positive integers A such that hA(σf ) > 0.(c) For any increasing sequence of positive integers A there exists a chaoticmap fA : [0, 1]→ [0, 1] such that hA(σfA) = 0.Proof. It is clear thatlim← ([0, 1], f) = {(x0, x1, . . . , xn, . . . ) : xi ∈ [a, b], f(xi) = xi−1} = lim← ([a, b], f).First of all we prove (a). Assume that f is chaotic. By Proposition 2.1, f |[a,b]is also chaotic. Applying Theorem 1.1 it follows that σf is chaotic. Conversely,suppose that σf is chaotic. Applying Theorem 1.1 it follows that f |[a,b] is chaotic.Proposition 2.1 proves that f is chaotic.Part (b). If σf is chaotic, then it follows by (a) that f is chaotic. Hence,by Proposition 2.1, f |[a,b] is chaotic. Applying Theorem 1.2 (a), there exists an208 J. S. CANOVASincreasing sequence of positive integers such that hA(f |[a,b]) > 0. Since f |[a,b]is surjective, by (2), hA(σf ) = hA(f |[a,b]) > 0. Now suppose that σf is non-chaotic. Assertion (a) states that f is non-chaotic. Applying Theorem 1.2 and(1), we conclude that hA(σf ) ≤ hA(f) = 0 for any increasing sequence of positiveintegers A.Part (c). Let A be an arbitrary sequence of positive integers. By Theo-rem 1.2(b), there exists a chaotic map fA : [0, 1] → [0, 1] such that hA(fA) = 0.Since fA is chaotic, by (a), σfA is also chaotic. By (2), hA(σfA) ≤ hA(fA) = 0,and the proof ends. 3. A CounterexampleAs usual, Z will stand for the set of integers, while if Z ⊂ Z then Zn (resp.Z∞) will denote the set of finite sequences of length n (resp. infinite sequences)of elements from Z. If θ ∈ Zn or α ∈ Z∞ then we will often describe themthrough their components as (θ1, θ2, . . . , θn) or (αi)∞i=1, respectively. The shiftmap σ : Z∞ → Z∞ is defined by σ((αi)∞i=1) = (αi+1)∞i=1. If θ ∈ Zn and ϑ ∈ Zm(with m ≤ ∞) then θ∗ϑ ∈ Zn+m (where n+∞ means∞) will denote the sequenceλ defined by λi = θi if 1 ≤ i ≤ n and λi = ϑi−n if i > n. In what follows wewill denote 0 = (0, 0, . . . , 0, . . . ) and 1 = (1, 1, . . . , 1, . . . ), while if α ∈ Z∞, thenα|n ∈ Zn is defined by α|n = (α1, α2, . . . , αn).This section is devoted to construct a chaotic map f on a compact metric spacefor which σf is non-chaotic. In order to do this we will need some informationconcerning so-called weakly unimodal maps of type 2∞. Recall briefly thedefinition. We say that a continuous map f : [0, 1] → [0, 1] is weakly unimodalif f(0) = f(1) = 0, it is non-constant and there is c ∈ (0, 1) such that f |(0,c) andf |(c,1) are monotone. The map f is said to be of type 2∞ if it has periodic pointsof period 2n for any n ≥ 0 but no other periods.Weakly unimodal maps of type 2∞ (briefly, w-maps) were studied in [4]. Inthat paper it was proved that for any w-map f it is possible to construct a family{Kα(f)}α∈Z∞ (or simply {Kα}α∈Z∞ if there is no ambiguity on f) of pairwisedisjoint (possibly degenerate) compact subintervals of [0, 1] satisfying the followingkey properties (P1)–(P4):(P1) The interval K0 contains all absolute maxima of f .(P2) Define in Z∞ the following total ordering: if α, β ∈ Z∞, α 6= β and k isthe first integer such that αk 6= βk, then α < β if either Card {1 ≤ i <k : αi ≤ 0} is even and αk < βk or Card {1 ≤ i < k : αi ≤ 0} is oddand βk < αk. Then α < β if and only if Kα < Kβ (that is, x < y for allx ∈ Kα, y ∈ Kβ).(P3) Let α ∈ Z∞, α 6= 0, and let k be the first integer such that αk 6= 0. Defineβ ∈ Z∞ by βi = 1 for 1 ≤ i ≤ k − 1, βk = 1− |αk| and βi = αi for i > kON SEQUENCE ENTROPY AND CHAOS 209Then f(Kα) = Kβ and f(K0) ⊂ K1.For any n and α ∈ Z∞, let Kα|n(f) (or just Kα|n) be the least interval includingall intervals Kβ , β ∈ Z∞, such that α|n = β|n. Then(P4) For any α ∈ Z∞, Kα =⋂∞n=1Kα|n .Additionally, for any fixed n it can be easily checked that the intervals Kθ, θ ∈Zn, are open and pairwise disjoint and (after replacing ∞ by n, 0 by (0, 0, . . . , n0)and 1 by (1, 1, . . . ,n1)), they also satisfy (P1)–(P3). Observe that if θ ∈ {−1, 0, 1}nand we put |θ| := (|θ1|, |θ2|, . . . , |θn|) then f2n(Kθ) ⊂ K|θ|; in particular,f2n(Kθ) ⊂ Kθ if θ ∈ {0, 1}n.In the rest of this section f˜ will denote a fixed w-map with the additionalproperty that α ∈ Z∞ implies Kα(f˜) is non-degenerate if and only if there is ann ≥ 0 such that σn(α) = 0. An example of such a map is constructed in [4];it is possible to show that the stunted tent map f˜(x) = min{1 − |2x − 1|, µ}(µ ≈ 0.8249 . . . ) from [7] is also a w-map with this property.Bd (Z), Cl (Z) and Int (Z) will respectively denote the boundary, the closureand the interior of Z.Now, we are ready to construct our counterexample. ConsiderX =⋃α∈{−1,0,1}∞Bd (Kα).Let us emphasize that Bd (Kα) consists of both endpoints of Kα if it is non-degenerate and of its only point if it is degenerate. Let f = f˜ |X : X → X be therestriction of the above-mentioned w-map f˜ to the set X. The following lemmashows that the above choices make sense.Lemma 3.1. X is a compact set and f : X → X is a well-defined continuousmap.Proof. SinceX = ∞⋂n=1⋃θ∈{−1,0,1}nCl (Kθ) \ ⋃α∈{−1,0,1}∞Int (Kα),by (P2) and (P4), X is compact.Recall that if 0 6= α ∈ {−1, 0, 1}∞ then f˜ carries the interval Kα onto Kβ with βdefined as in (P3) (and hence belonging to {−1, 0, 1}∞). Moreover, f˜ is monotoneon Kα because of (P1). So it maps the endpoints of Kα onto the endpoints of Kβ .Similarly, since K1 is degenerate both endpoints of K0 are mapped onto its onlypoint. The conclusion is that f(X) ⊂ X and the map f : X → X is well–defined(and it is clearly continuous). Let X1 =⋃α∈{0,1}∞ Bd (Kα). Note that, by (P3),⋂n≥0 fn(X) = X1. Let ussee that f is chaotic while f |X1 is non–chaotic.210 J. S. CANOVASTheorem 3.2. f |X1 is non–chaotic and hence σf is non–chaotic.Proof. Let x, y ∈ X1 with x ∈ Kα and y ∈ Kβ for some α 6= β, α, β ∈ {0, 1}∞.We will see that there exists a positive real number M satisfyinglim infi→∞|f i(x)− f i(y)| ≥M,and hence x and y cannot belong to the same scrambled set D. This proves thatCard(D) ≤ 2 for each scrambled set D of f |X1 and so f |X1 is non–chaotic.Let j be the first positive integer satisfying αj 6= βj . Suppose, for example,that αj = 0 and βj = 1. For any θ ∈ {0, 1}j−1 consider the closed interval Aθ∗0satisfying Kθ∗1 < Aθ∗0 < Kθ∗0 or Kθ∗0 < Aθ∗0 < Kθ∗1, and let M = min{|Aθ∗0| :θ ∈ {0, 1}j−1} > 0. By (P3) and (P2), f i(x) ∈ Kθ∗1 and f i(y) ∈ Kθ∗0 or viceversafor all i ∈ N and for all θ ∈ {0, 1}j−1. This shows that|f i(x)− f i(y)| ≥Mwhich concludes the proof. For any α ∈ {−1, 0, 1}∞ let τ(α|n) =∑ni=1 |αi|2i−1 for all n ∈ N.Theorem 3.3. f : X → X is chaotic.Proof. Define on {−1, 1}∞ the following relation: α ∼ β if and only if thereexists a positive integer k such that σk(α) = σk(β). Obviously ∼ is an equivalencerelation. Moreover, for α ∈ {−1, 1}∞ the class of α is given by[α] =∞⋃k=0{σ−k(σk(α))} ∩ {−1, 1}∞.Since {−1, 1}∞ is uncountable and [α] is countable for all α ∈ {−1, 1}∞, the setcontaining all the equivalence classes {−1, 1}∞/ ∼ is uncountable.Let A be a set containing one and only one representative α ∈ [α] for all[α] ∈ {−1, 1}∞/ ∼, and let D be the set containing exactly one x ∈ X ∩Kα forall α ∈ A. We claim that D is a scrambled set for f . In order to see this takex, y ∈ D, x ∈ Kα and y ∈ Kβ with α, β ∈ A, α 6= β. Then there exists anincreasing sequence of positive integers (ki)∞i=0 satisfying αki 6= βki and αj = βjif j 6= ki for all i ∈ N. Note that τ(α|n) = τ(β|n) for all n ∈ N. Suppose, forexample, that αki = 1 and βki = −1 for some i. Then, by (P3),fτ(α|ki )(x) ∈ K(0,0,...,ki−10 ,1)∗σki (α)and fτ(α|ki )(y) ∈ K(0,0,...,ki−10 ,−1)∗σki (β).By (P2), |fτ(α|ki )(x)− fτ(α|ki )(y)| ≥ |K0| and thenlim supn→∞|fn(x)− fn(y)| ≥ lim supi→∞|fτ(α|ki )(x)− fτ(α|ki )(y)| ≥ |K0|.ON SEQUENCE ENTROPY AND CHAOS 211Let now j 6= ki for all i ∈ N, and suppose that αj = βj = 1 (the case αj = −1is analogous). Then fτ(α|j)(y), fτ(α|j)(x) ∈ K(0,0,...,0,j1). For any n ∈ N let K+0|nand K−0|n be the right and left side components of K0 \K0|n . Applying (P4), forany ε > 0 there exists a positive integer nε such that max{|K+0|n |, |K−0|n |} < ε forall n ≥ nε. By (P2) and (P3), K(0,0,...,0,j1)⊂ K−0|j or K(0,0,...,0,j1) ⊂ K+0|j , and sofor j ≥ nε we conclude that |fτ(α|j)(y)− fτ(α|j)(x)| < ε. This proves thatlim infn→∞ |fn(y)− fn(x)| = 0,and the proof concludes. Acknowledgement. This paper has been partially supported by the grantPB/2/FS/97 (Fundacio´n Se´neca, Comunidad Auto´noma de Murcia).I wish to thank the referee for the proof of Proposition 2.1 and the commentsthat helped me to improve the paper.References1. Franzova´ N. and Smı´tal J., Positive sequence topological entropy characterizes chaotic maps,Proc. Amer. Math. Soc. 112 (1991), 1083–1086.2. Goodman T. N. T., Topological sequence entropy, Proc. London Math. Soc. 29 (1974),331–350.3. Gu Rongbao, Topological entropy and chaos of shift maps on the inverse limits spaces,J. Wuhan Univ. (Natural Science Edition) 41 (1995), 22–26. (Chinese)4. Jime´nez Lo´pez V., An explicit description of all scrambled sets of weakly unimodal functionsof type 2∞, Real. Anal. Exch. 21 (1995/1996), 1–26.5. Hric R., Topological sequence entropy for maps of the interval, to appear in Proc. Amer.Math. Soc.6. Li Shihai, Dynamical properties of the shift maps on the inverse limit spaces, Ergod. Th. andDynam. Sys 12 (1992 95–108).7. Misiurewicz M. and Smı´tal J., Smooth chaotic functions with zero topological entropy, Ergod.Th. and Dynam. Sys. 8 (1988), 421–424.8. Paganoni L. and Santambrogio P., Chaos and sequence topological entropy for triangularmaps, quaderno n. 59/1996, Universta´ degli studi di Milano.J. S. Canovas, Departamento de Matema´tica Aplicada, Universidad Polite´cnica de Cartagena,Paseo de Alfonso XIII, 34–36, 30203 Cartagena, Spain; e-mail : canovas@plc.um.es

On topological sequence entropy and chaotic maps on inverse limit spaces

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