summary:A continuous map $f$ of the interval is chaotic iff there is an increasing sequence of nonnegative integers $T$ such that the topological sequence entropy of $f$ relative to $T$, $h_T(f)$, is positive ([FS]). On the other hand, for any increasing sequence of nonnegative integers $T$ there is a chaotic map $f$ of the interval such that $h_T(f)=0$ ([H]). We prove that the same results hold for maps of the circle. We also prove some preliminary results concerning topological sequence entropy for maps of general compact metric spaces

Hric, Roman

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Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeRoman HricTopological sequence entropy for maps of the circleCommentationes Mathematicae Universitatis Carolinae, Vol. 41 (2000), No. 1, 53--59Persistent URL: http://dml.cz/dmlcz/119140Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 2000Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 41,1 (2000)53–59 53Topological sequence entropy for maps of the circleRoman HricAbstract. A continuous map f of the interval is chaotic iff there is an increasing sequenceof nonnegative integers T such that the topological sequence entropy of f relative to T ,hT (f), is positive ([FS]). On the other hand, for any increasing sequence of nonnegativeintegers T there is a chaotic map f of the interval such that hT (f) = 0 ([H]). We provethat the same results hold for maps of the circle. We also prove some preliminary resultsconcerning topological sequence entropy for maps of general compact metric spaces.Keywords: chaotic map, circle map, topological sequence entropyClassification: Primary 26A18, 54H20, 58F13IntroductionLet (X, ρ) be a compact metric space; denote by C(X) the space of all con-tinuous maps of this space into itself. We will pay a special attention to thecase when X is the circle S = {z ∈ C; |z| = 1}; the metric on S is given by‖x, y‖ = dist(Π−1x,Π−1y) where Π denotes the natural projection of the realline R onto S, i.e., Π(x) = e2πix. By N we denote the set of all positive integers.If T = (ti)∞i=1 is an arbitrary sequence of nonnegative integers then the (T, f, n)-trajectory of x ∈ X is the sequence (f tix)ni=1. The set of all periodic points of fis denoted by Per(f) and the set of periods of all periodic points of f by P (f).A set A ⊆ X is called a retract of X if there is a map r : X → A such thatr(a) = a for every a ∈ A.Definition. Let (X, ρ) be a compact metric space. A map f ∈ C(X) is said tobe chaotic if there are points x, y ∈ X such thatlim supi→∞ρ(f ix, f iy) > 0,lim infi→∞ρ(f ix, f iy) = 0.(The set {x, y} is called a scrambled set.) A map is called nonchaotic if it is notchaotic.Remark. This definition of a chaotic map is equivalent to the original one by Liand Yorke in [LY] for maps of the interval (see [KuS]) and for maps of the circle(see [Ku]).The author has been supported by the Slovak grant agency, grant number 1/4015/97.54 R.HricDefinition. Let (X, ρ) be a compact metric space, f ∈ C(X) and T = (ti)∞i=1be an increasing sequence of nonnegative integers. We say that a set A ⊆ X(T, f, ε, n)-spans a set B ⊆ X if for any x ∈ B there is y ∈ A such thatρ(f tix, f tiy) < ε for all 1 ≤ i ≤ n. (We also say that the point y spans thepoint x.)Definition ([G]). Let (X, ρ) be a compact metric space, f ∈ C(X) and T =(ti)∞i=1 be an increasing sequence of nonnegative integers.A set A ⊆ X is said to be (T, f, ε, n)-separated if for any x, y ∈ A, x 6= y thereis an index i, 1 ≤ i ≤ n, such that ρ(f tix, f tiy) > ε. Let Sep(T, f, ε, n) denotethe largest of cardinalities of all (T, f, ε, n)-separated sets. PutSep(T, f) = limε→0lim supn→∞1nlog Sep(T, f, ε, n).A subset of X is said to be a (T, f, ε, n)-span if it (T, f, ε, n)-spans X . LetSpan(T, f, ε, n) denote the smallest of cardinalities of all (T, f, ε, n)-spans. PutSpan(T, f) = limε→0lim supn→∞1nlog Span(T, f, ε, n).Then Sep(T, f) = Span(T, f) (see [G]) and we define the topological sequenceentropy of f relative to T , hT (f), to be Sep(T, f).Remark. If ti = i − 1, i = 1, 2, . . . then hT (f) is the topological entropy h(f)of f . Topological sequence entropy can be viewed as the topological entropy ofthe nonautonomous dynamical system given on the space X by the sequence ofmaps f t1 , f t2−t1 , f t3−t2 , . . . (see [KS]).In [FS] Franzová and Smı́tal proved that a continuous map f of the intervalis chaotic if and only if there is an increasing sequence of nonnegative integers Tsuch that hT (f) > 0. A natural question arose whether there is some universalsequence which characterizes chaos. This is not the case as it was proved in [H]— for any increasing sequence of nonnegative integers T there is a chaotic mapf with hT (f) = 0. The main aim of this paper is to prove the same results formaps of the circle.Theorem 1. A map f ∈ C(S) is chaotic if and only if there is an increasingsequence of nonnegative integers T such that hT (f) > 0.Remark. Theorem 1 does not hold in general, even for triangular maps of thesquare. There is a nonchaotic triangular map with positive topological sequenceentropy relative to a suitable sequence (see [FPS, Theorem 2]) and, on the otherhand, there is a chaotic triangular map with zero topological sequence entropyrelative to any sequence (see [FPS, Theorem 3]).Theorem 2. LetX be a compact metric space containing a homeomorphic imageof an interval and let T be an increasing sequence of nonnegative integers. Thenthere is a chaotic map f ∈ C(X) such that hT (f) = 0.Remark. The analysis of the proof of Theorem 6 in [H] shows that Theorem 2holds also when X is a Cantor set.Topological sequence entropy for maps of the circle 55Corollary 3. Let T be an increasing sequence of nonnegative integers. Thenthere is a chaotic map f ∈ C(S) such that hT (f) = 0.Preliminary resultsLet (X, ρ) and (Y, σ) be compact metric spaces, f ∈ C(X), g ∈ C(Y ), and letπ : X → Y be a continuous map such that the diagramXf−−−−→ XπyyπYg−−−−→ Ycommutes. In this situation we have the followingLemma 4. Let T be an increasing sequence of nonnegative integers. Then(i) if π is injective then hT (f) ≤ hT (g);(ii) if π is surjective then hT (f) ≥ hT (g);(iii) if π is bijective then hT (f) = hT (g).Proof:(ii) and (iii). See [G, p. 332].(i). We have that π is a homeomorphism between X and πX . Thus, by (iii),hT (f) = hT (g|πX). Now let E ⊆ πX be (T, g|πX , ε, n)-separated. Trivially, it isalso (T, g, ε, n)-separated which gives hT (g|πX ) ≤ hT (g). It is known that some of the properties of topological entropy are not satisfiedby topological sequence entropy. For example, contrary to the formula h(fk) =k · h(f), an analogous formula for topological sequence entropy does not hold —it is even possible that hS(f) < hT (f) for a subsequence S of T ([L]). In this casethe following result can be useful.Theorem 5. Let (X, ρ) be a compact metric space, f ∈ C(X), T be an increasingsequence of nonnegative integers and k be a positive integer. Then there is anincreasing sequence of nonnegative integers S such that hS(fk) ≥ hT (f).Proof: SinceX is compact, f, f2, . . . , fk−1 are equicontinuous, i.e., for any ε > 0there is δ = δ(ε) > 0 such that if x, y ∈ X and ρ(x, y) ≤ δ then ρ(f ix, f iy) < εfor i = 1, . . . , k − 1. We may assume that δ ≤ ε.Let T = (ti)∞i=1. Define S = (si)∞i=1 as follows. Put s1 =[t1k](where [·] standsfor the integer part) and for anym let sm+1 will be the first[tik]greater than sm.Let E ⊆ X be an (T, f, ε, n)-separated set. We are going to show that E isa (S, fk, δ, m)-separated set where m is such that sm =[tnk]. To this end letx, y ∈ E, x 6= y. Then for some i ∈ {1, 2, . . . , n}, ρ(f tix, f tiy) > ε. Take j withsj =[tik]. Then j ≤ m and from the definition of δ we have ρ(fk·sjx, fk·sjy) > δ.Thus E is an (S, fk, δ, m)-separated set. From this we have Sep(T, f, ε, n) ≤56 R.HricSep(S, fk, δ, m). Now, hT (f) = limε→0lim supn→∞1n log Sep(T, f, ε, n) ≤limδ→0lim supn→∞1n log Sep(S, fk, δ, m) ≤ limδ→0lim supm→∞1m log Sep(S, fk, δ, m) = hS(fk).Corollary 6. Let X be a compact metric space, f ∈ C(X) and k be a positiveinteger. Then the following two conditions are equivalent:(i) there is an increasing sequence T of nonnegative integers such thathT (f) > 0;(ii) there is an increasing sequence T of nonnegative integers such thathT (fk) > 0.In the sequel we will discuss the space of maps of the circle. The space C(S)can be decomposed into the following classes (see [ALM, Chapter 3], cf. also [Ku,p. 384]):C1 = {f ∈ C(S); f has no periodic point};C2 = {f ∈ C(S); P (fn) = {1} or P (fn) = {1, 2, 22, . . . } for some n ∈ N};C3 = {f ∈ C(S); P (fn) = N for some n ∈ N}.According to this we will distinguish three different cases.Maps withou t periodic pointsIn all of this section we assume f ∈ C(S) to have no periodic point. We aregoing to show that Theorem 1 holds for such maps. Since, by [Ku, Theorem B], fis not chaotic, we need only to show that hT (f) = 0 for any increasing sequence T .So fix T . If f is a homeomorphism then hT (f) = 0 by [KS, TheoremD]. Otherwise,by [AK, Theorem 1 and Theorem 2], there is a nowhere dense perfect set E whichis the only ω-limit set of f , all (closed) contiguous intervals are wandering, thepreimage of any contiguous interval is a contiguous interval, the image of anycontiguous interval is either a contiguous interval or a point from E. Moreover,f |E is monotone. By linear extension of f |E we obtain a monotone map g ∈ C(S).By [KS, Theorem D], hT (g) = 0. By Lemma 4(i), hT (f |E) ≤ hT (g). Hence,lim supn→∞1n log Span(T, f |E , ε, n) = 0 for any ε > 0.Now fix an arbitrary ε > 0. We are going to estimate Span(T, f, ε, n). LetI1, . . . , Ik be all contiguous intervals longer thanε2 . Let A be a (T, f |E ,ε2 , n)-span.Take any point x whose (T, f, n)-trajectory lies in S \⋃ki=1 Ii. If x ∈ E then x is(T, f, ε, n)-spanned by A. For x /∈ E put y to be an endpoint of the contiguousinterval which contains x. Then ‖f tix, f tiy‖ ≤ ε2 for all 1 ≤ i ≤ n. Since y ∈ Eis (T, f, ε2 , n)-spanned by a point z ∈ A, the set A obviously (T, f, ε, n)-spans allsuch points x.So it remains to consider those points whose (T, f, n)-trajectories meet⋃ki=1 Ii.Fix N ∈ N such thatN > 1ε . We are going to show that there is a set of cardinalityat most n · k · Nk which (T, f, ε, n)-spans all considered points. It is sufficientto show that there is a set with cardinality at most Nk which (T, f, ε, n)-spansTopological sequence entropy for maps of the circle 57the set I(ti, Ij) = {x ∈ S; ftix ∈ Ij} (for fixed 1 ≤ i ≤ n and 1 ≤ j ≤ k).First, it is obvious that I(ti, Ij) is a contiguous interval. Consider its (T, f, n)-trajectory (f t1I(ti, Ij), . . . , ftnI(ti, Ij)). Each element in this trajectory is eithera contiguous interval or a point from E. At most k of them have lengths greaterthan or equal to ε— cut each of such elements to N segments shorter than ε. Allthe other elements of the trajectory will be considered to be segments themselves.To each x ∈ I(ti, Ij) assign its code — the sequence (S1(x), . . . , Sn(x)) whereSl(x) is the segment containing ftlx. We have at most Nk different codes and allpoints with the same code can be (T, f, ε, n)-spanned by one point.From what has been said above we see that Span(T, f, ε, n) ≤ Span(T, f |E ,ε2 , n)+n·k ·Nk which finishes the proof of Theorem 1 for maps without periodic points.Maps with periodic pointsWe will first deal with the case C2. We know that for any n ∈ N f is chaoticif and only if fn is chaotic. Taking into account Corollary 6 we can without lossof generality assume that P (f) = {1} or P (f) = {1, 2, 22, . . . }. Since f has afixed point, by [Ku, Lemma 2.5] there is a lifting F and an F -invariant compactinterval J longer than 1. In the following discussion of the case C2 we will write Fand Π instead of F |J and Π|J , respectively, as in the next commutative diagramJF−−−−→ JΠyyΠSf−−−−→ SNote that if x, y ∈ J then ‖Πx,Πy‖ ≤ |x−y| with the equality whenever |x−y| ≤12 .Lemma 7. F is chaotic if and only if f is chaotic.Proof: Let F be chaotic. Then there are two points u, v ∈ J such thatlim supi→∞|F iu − F iv| = γ > 0;(1)lim infi→∞|F iu − F iv| = 0.(2)We claim that Πu,Πv form a scrambled set for f . From (2),lim infi→∞ ‖fiΠu, f iΠv‖ = 0. Now put η = min{γ, 12}. Take 0 < δ < η suchthat |x − y| < δ implies |Fx − Fy| < η. From this and (1) and (2) we havethat |F iu − F iv| ∈ [δ, η] infinitely many times. Since η ≤ 12 , the same holds for‖f iΠu, f iΠv‖. Thus lim supi→∞ ‖fiΠu, f iΠv‖ ≥ δ > 0.Let F be nonchaotic. Then by [JS, Theorem 3] every trajectory of F is ap-proximable by cycles, i.e. for any ε > 0 and any x ∈ J there is some periodicpoint p ∈ Per(F ) such that(3) |F ix − F ip| < ε for all i = 0, 1, 2, . . . .58 R.HricFix any z ∈ S. Take any of its preimages x ∈ Π−1z. Let ε > 0 be arbitrary,p ∈ Per(F ) such that (3) is satisfied. Clearly, Πp is a periodic point for f and‖f iz, f iΠp‖ < ε for all i = 0, 1, 2, . . . . Thus f is not chaotic by [Ku, Theorem A].Lemma 8. Let F be chaotic. Then there is an increasing sequence T such thathT (f) > 0.Proof: If F has a periodic point of period k · 2m where k > 1 is odd then, bySharkovsky theorem, it has also a periodic point of period k′ · 2m where k′ >diamJ + 1 is odd. Since Π|J is at most [diamJ ] + 1 to one, f has a periodicpoint of period k′′ · 2m′where k′′ > 1 is odd. This is a contradiction since P (f)is {1} or {1, 2, 22, . . . }. So F is of type 2∞, chaotic. By [S] there is an orbit ofperiodic intervals of period p > diamJ such that F p is chaotic on each of them.At least one interval K in this orbit is shorter than 1. Then Π|K is injectiveand so F p|K is topologically conjugate with fp|ΠK . By [FS, Theorem] there isan increasing sequence of nonnegative integers S such that hS(Fp|K) > 0. Sincehp·S(f) = hS(fp) it is sufficient to use Lemma 4(iii) and (i) to get hp·S(f) ≥hS(fp|πK) = hS(Fp|K) > 0. We are going to show that Theorem 1 holds for maps from the class C2. Letf ∈ C2 be chaotic. Then we obtain the required result using Lemma 7 andLemma 8.Now let f ∈ C2 and let there be an increasing sequence of nonnegative integersT such that hT (f) > 0. Theorem 4(ii) then implies that hT (F ) > 0 where F hasthe same meaning as above. By [FS, Theorem] F is chaotic. Lemma 7 finishesthe proof.Finally we will discuss the situation for maps from the remaining class C3. Solet P (fn) = N for some n. By [BC, Theorem IX.28(i) and (ii)] we have thath(fn) is positive and so is h(f). By the same theorem, conditions (ii) and (iii),we have that fm·n is strictly turbulent for a suitable m ∈ N which implies thatf is chaotic for the same reason as in the interval case. This finishes the proof ofTheorem 1.Proof of Theorem 2The space X contains a homeomorphic image J of the interval [0, 1]. The setJ is a retract of X by [HY, Theorem 2-34]. Let r : X → J be a correspondingretraction. By [H, Theorem 6] there is a chaotic onto map g ∈ C([0, 1]) suchthat hT (g) = 0. Let g̃ ∈ C(J) be a map topologically conjugate with g. Definef ∈ C(X) by f = g̃ ◦ r. Since⋂∞i=0 fiX = fX = J , we have that hT (f) =hT (f |J ) = 0 by [BCJ, Proposition 1].Acknowledgment. The author thanks Professor L’ubomı́r Snoha for contribut-ing ideas useful for this paper and for his helpful comments during the last stageof its preparation.Topological sequence entropy for maps of the circle 59References[ALM] Alsedà L., Llibre J., Misiurewicz M., Combinatorial Dynamics and Entropy in Dimen-sion One, World Scientific Publ., Singapore, 1993.[AK] Auslander J., Katznelson Y., Continuous maps of the circle without periodic points,Israel. J. Math. 32 (1979), 375–381.[BCJ] Balibrea F., Cánovas J.S., Jiménez López V., Commutativity and noncommutativity oftopological sequence entropy, preprint.[BC] Block L.S., Coppel W.A., Dynamics in One Dimension, Lecture Notes in Math., vol.1513, Springer, Berlin, 1992.[FS] Franzová N., Smı́tal J., Positive sequence entropy characterizes chaotic maps, Proc.Amer. Math. Soc. 112 (1991), 1083–1086.[G] Goodman T.N.T., Topological sequence entropy, Proc. London Math. Soc. 29 (1974),331–350.[HY] Hocking J.G., Young G.S., Topology, Dover, New York, 1988.[H] Hric R., Topological sequence entropy for maps of the interval, Proc. Amer. Math. Soc.127 (1999), 2045–2052.[JS] Janková K., Smı́tal J., A characterization of chaos, Bull. Austral. Math. Soc. 34 (1986),283–292.[KS] Kolyada S., Snoha L’., Topological entropy of nonautonomous dynamical systems, Ran-dom and Comp. Dynamics 4 (1996), 205–233.[Ku] Kuchta M., Characterization of chaos for continuous maps of the circle, Comment.Math. Univ. Carolinae 31 (1990), 383–390.[KuS] Kuchta M., Smı́tal J., Two point scrambled set implies chaos, European Conference onIteration Theory ECIT’87, World Sci. Publishing Co., Singapore.[L] Lemańczyk M., The sequence entropy for Morse shifts and some counterexamples, Stu-dia Math. 52 (1985), 221–241.[LY] Li T Y., Yorke J.A., Period three implies chaos, Amer. Math. Monthly 82 (1975), 985–992.[S] Smı́tal J., Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297(1986), 269–281.Department of Mathematics, Faculty of Natural Sciences, Matej Bel University,Tajovského 40, SK–974 01 Banská Bystrica, Slovak RepublicE-mail : hric@fpv.umb.sk(Received March 31, 1999)

Topological sequence entropy for maps of the circle

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Topological sequence entropy for maps of the circle

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Institute of Mathematics AS CR, v. v. i.