13.dvi @ Applied General TopologyUniversidad Polit�ecnica de ValenciaVolume 2, No. 1, 2001pp. 1 - 7 On topological sequence entropy of circlemapsJos�e S. C�anovas�Abstract. We classify completely continuous circle maps fromthe point of view of topological sequence entropy. This improvesa result of Roman Hric.2000 AMS Classi�cation: 37B40, 37E10.Keywords: Topological sequence entropy, circle maps, chaos in the sense ofLi{Yorke. 1. IntroductionLet (X;d) be a compact metric space and let f : X ! X be a continuousmap. Denote by C(X;X) the set of continuous maps f : X ! X. (X;f) iscalled a discrete dynamical system. The map f is said chaotic in the sense ofLi{Yorke (or simply chaotic) if there is an uncountable set S � X such thatfor any x;y 2 S, x 6= y, it holds that(1.1) lim infn!1 d(fn(x);fn(y)) = 0;and(1.2) lim supn!1 d(fn(x);fn(y)) > 0:S is said a scrambled set of f (see [10]). When f is chaotic we say that (X;f)is chaotic.The notion of chaos plays an special role in the setting of discrete dynamicalsystems. So, some topological invariants have been porposed to give a chara-terization of chaos. Maybe, the most important topological invariant in thissetting is the topological entropy (see [1]). When X = [a;b], a;b 2 R, it iswell{known that positive topological entropy implies that f is chaotic, whilethe converse result is false (see [12]).So, in order to characterize chaotic interval maps we need an extension oftopological entropy called topological sequence entropy (see [7]). Given an�This paper has been partially supported by the grant D.G.I.C.Y.T. PB98-0374-C03-01. 2 Jos�e S. C�anovasincreasing sequence of positive integers A = (ai)1i=0, a number hA(f) can beassociated to each f 2 C(X;X). This number is also a topological invariant.Then, de�ning h1(f) = supA hA(f), chaotic interval maps can be characterizedby the following result.Theorem 1.1. Let f 2 C([a;b]; [a;b]). Then(a) f is non{chaotic i� h1(f) = 0.(b) f is chaotic with zero topological entropy i� h1(f) = log 2.(c) f is chaotic with positive topological entropy i� h1(f) = 1.Theorem 1.1 establishes a complete classi�cation of maps from the point ofview of topological sequence entropy. The part (a) was proved by Franzov�aand Sm��tal in [6]. (a) provides that any chaotic map holds h1(f) > 0. In [3]was proved (b) and (c) in case of piecewise monotonic maps. This result wasextended to the general case in [4].Following [7], a map f 2 C(X;X) is said null if h1(f) = 0. f is said boundedif h1(f) < 1 and unbounded if h1(f) = 1. In the general case, it is unknownwhen a continuous map is null, bounded or unbounded. It is easy to see thatwhen f is stable in the Lyapunov sense (f has equicontinuous powers) the mapis null (see [7]). Theorem 1.1 establishes a characterization of null, boundedand unbounded continuous interval maps.The aim of this paper is to prove Theorem 1.1 in the setting of continuouscircle maps. This will provide a classi�cation of unbounded, bounded and nullcontinuous circle maps. Before starting with this classi�cation, let us point outthat for any f 2 C(S1;S1), Hric proved in [8] that it is non{chaotic i� h1(f) =0, which classi�es chaotic circle maps from the point of view of topologicalsequence entropy. 2. PreliminariesLet (X;d) be a compact metric space and let f : X ! X be a continuousmap. Denote by C(X;X) the set of continuous maps f : X ! X. Let f0 be theidentity on X, f1 := f and fn = f � fn�1 for all n � 1. Consider an increasingsequence of positive integers A = (ai)1i=1 and let Y � X and " > 0. We saythat a subset E � Y is (A;";n;Y;f){separated if for any x;y 2 E, x 6= y, thereis an i 2 f1;2; :::;ng such that d(fai(x);fai(y)) > ". Denote by sn(A;";Y;f)the cardinality of any maximal (A;";n;Y;f){separated set. De�ne(2.3) s(A;";Y;f) := lim supn!1 1n log sn(A;";Y;f):It is clear from the de�nition that if Y1 � Y2 � X, then(2.4) s(A;";Y1;f) � s(A;";Y2;f):Let(2.5) hA(f;Y ) := lim"!0 s(A;";Y;f):The topological sequence entropy of f respect to the sequence A is de�ned by(2.6) hA(f) := hA(f;X): On t. s. e. of circle maps 3When A = (i)1i=0, we receive the classical de�nition of topological entropy (see[1]).Finally, let(2.7) h1(f;Y ) := supA hA(f;Y )and(2.8) h1(f) := supA hA(f):An x 2 X is periodic if there is an n 2 N such that fn(x) = x. The smallestpositive integer holding this condition is called the period of x. The set ofperiods of f, P(f), is de�ned byP(f) := fn 2 N : 9x 2 X periodic point of periodng:3. Results on topological sequence entropyIn this section we prove some useful results concerning topological sequenceentropy of continuous maps de�ned on arbitrary compact metric spaces.Proposition 3.1. Let f 2 C(X;X). For all n 2 N it holds that h1(fn) =h1(f).Proof. First, we prove that h1(fn) � h1(f). In order to see this, let A =(ai)1i=1 be an increasing sequence of positive integers and de�ne nA = (nai)1i=1.Then it is straightforward to see that hA(fn) = hnA(f) and henceh1(fn) = supA hA(fn) = supA hnA(f)� supB hB(f) = h1(f):Now, we prove the converse inequality. Let A be an increasing sequence ofpostive integers. By [8], there is another sequence B = B(A) such that hA(f) �hB(fn). Then h1(f) = supA hA(f) � supA hB(A)(fn)� supA hA(fn) = h1(fn);which ends the proof. �Corollary 3.2. Under the conditions of Proposition 3.1, the following state-ments hold:(a) f is null i� fn is null for all n 2 N.(b) f is bounded i� fn is bounded for all n 2 N.(c) f is unbounded i� fn is unbounded for all n 2 N.Proposition 3.3. Let f 2 C(X;X) have positive topological entropy. Thenh1(f) = 1. 4 Jos�e S. C�anovasProof. Since h(f) > 0 it follows by [7] that for any increasing sequence ofpositive integers A = (ai)1i=1, hA(f) = K(A)h(f), where(3.9) K(A) = limk!1 lim supn!1 1nCardfai;ai + 1; :::;ai + k : 1 � i � ng:Taking A = (2i)1i=1 it holds that K(A) = 1 and hence hA(f) = 1. �Proposition 3.4. Let (X;d) and (Y;e) be compact metric spaces and let f :X ! X and g : Y ! Y be continuous maps. Let � : X ! Y be continuous andsurjective such that � � f = g � �. Let A be an increasing sequence of positiveintegers A and let Y1 � Y . Then, for any " > 0 there is a � > 0 such that(3.10) s(A;�;��1(Y1);f) � s(A;";Y1;g):In particular, h1(f) � h1(g).Proof. Let E � Y1 be a maximal subset (A;n;";Y1;g){separated. Let F ���1(Y1) be a set containing exactly one element from ��1(y) for all y 2 E.We claim that F is an (A;n;�;��1(Y1);f){separated subset for some � > 0.Assume the contrary. Since � is uniformly continuous, there is a � = �(") > 0such that d(x1;x2) < �, x1;x2 2 X, implies e(�(x1);�(x2)) < ". Now letx1;x2 2 F be such that(3.11) d(fai(x1);fai(x2)) < �for all i 2 f1;2; :::;ng. Let y1;y2 2 E be such that �(xj) = yj for j = 1;2.Then, for all i 2 f1;2; :::;ng we have thate(gai (y1);gai(y2)) = e(gai (�(x1));gai(�(y2)))= e(� � fai(x1);� � fai(y2)) � ";which leads us to a contradiction. Then sn(A;�;��1(Y1);f) � sn(A;";Y1;f)and hence s(A;�;��1(Y1);f) = lim supn!1 1n log sn(A;�;��1(Y1);f)� lim supn!1 1n log sn(A;";Y1;g)= s(A;";Y1;g);which ends the proof. �Under the conditions of Proposition 3.4, if � is an homemorphism, then fand g are said to be conjugate. ThenCorollary 3.5. Under the conditions of Proposition 3.4, if f and g are conju-gate, then h1(f) = h1(g). On t. s. e. of circle maps 54. Main resultsIn the sequel we will discuss the space of continuous circle maps denoted byC(S1;S1). Let f 2 C(S1;S1) and let l : R ! S1 be de�ned by l(x) = exp(2�ix)for all x 2 R. Then, there are a countable number of continuous maps F : R !R such that l � F = f � l. An F holding this condition is called a lifting of f.If eF is another lifting of f, then(4.12) eF � F = k 2 N:By jJj we denote the length of an interval J � R.Theorem 4.1. Let f 2 C(S1;S1). Then(a) f is non{chaotic i� h1(f) = 0:(b) f is chaotic with zero topological entropy i� h1(f) = log 2:(c) f is chaotic with positive topological entropy i� h1(f) = 1:Proof. According to Chapter 3 from [2], C(S1;S1) can be decomposed into thefollowing classes:(4.13) C1 = ff 2 C(S1;S1) : f has no periodic pointsg;(4.14)C2 = ff 2 C(S1;S1) : P(fn) = f1g or P(fn) = f1;2;22; :::g for some n 2 Ng;(4.15) C3 = ff 2 C(S1;S1) : P(fn) = N for some n 2 Ng:According to [8], any f 2 C1 is non{chaotic and holds that h1(f) = 0. Letf 2 C3. Again by [8], it holds that f is chaotic and h(f) > 0. Then, byProposition 3.3 we have that h1(f) = 1. So, we must consider only mapsfrom C2.Let f 2 C2 and let n 2 N be such that P(fn) = f1g or P(fn) = f1;2;22; :::g.It is well{known that f is chaotic i� fn is chaotic. So, applying Proposition 3.1,it is not restrictive to assume that n = 1. Since f has a �xed point, by Lemma2.5 from [9], there is a lifting F : R ! R and there is a compact interval J, withjJj > 1, such that F(J) = J. For the rest of the proof call l = ljJ. First assumethat f is non{chaotic. Then by [8] it holds that h1(f) = 0. Secondly, assumethat f is chaotic. Hence F is also chaotic (see [8]) and has zero topologicalentropy (see [12]). By Proposition 3.4, for any " > 0 there is a � = �(") > 0such that(4.16) s(A;";S1;f) � s(A;�;l�1(S1);F) = s(A;�;J;F):On the other hand, by [4], there is a compact interval Ji � J, holding thatF2i(Ji) = Ji such that(4.17) s(A;�;J;F) � s(A;�=6;[2ij=1Fj(Ji);F) � log 2:By (4.16) and (4.17) we conclude that(4.18) s(A;";S1;f) � log 2:Since " was arbitrary chosen, we obtain(4.19) hA(f) � log 2; 6 Jos�e S. C�anovasand(4.20) h1(f) = supA hA(f) � log 2:Now, we prove the converse inequality. By [12], there is a compact intervalJi, with jJij < 1 and F2i(Ji) = Ji such that F2ijJi is chaotic. By [6], there isan increasing sequence of positive integers B such that s(B;";Ji;F2i) � log 2for a suitable " > 0. Since ljJi : Ji ! l(Ji) is an homemorphism, we can applyProposition 3.4 to ljJi = l to obtain a � > 0 such that(4.21) s(A;�;l(Ji);f2i) � s(A;";Ji;F2i) � log 2:Hence(4.22) h1(f) � h2iA(f) = hA(f2i) � s(A;�;l(Ji);f2i) � log 2;which concludes the proof. �Remark 4.2. When two{dimensional maps are concerned, Theorems 1.1 and4.1 are false in general. More precisely, in [11] and [5] a chaotic map F 2C([0;1]2; [0;1]2) with h1(F) = 0 and a non chaotic map G 2 C([0;1]2; [0;1]2)holding h1(G) > 0 have been constructed. It seems that the dimension of thespace X plays a special role in Theorems 1.1 and 4.1. We conjecture thatTheorem 1.1 remains true for continuous maps de�ned on �nite graphs, that is,in the special setting of one{dimensional dynamics.References[1] R. L. Adler, A. G. 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Carolin. 41(2000), 53{59.[9] M. Kuchta, Characterization of chaos for continuous maps of the circle, Comment. Math. Univ.Carolinae 31 (1990), 383{390.[10] T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985{992.[11] L. Paganoni and P. Santambrogio, Chaos and sequence topological entropy for triangular maps,Grazer. Math. Ber. 339 (1999), 279{290.[12] J. Sm��tal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986),269{282. Received July 2000 On t. s. e. of circle maps 7Revised version October 2000 Jos�e S. C�anovasDep. de Matem�atica Aplicada y Estad��sticaUniversidad Polit�ecnica de Cartagena30203 CartagenaSpainE-mail address: Jose.Canovas@upct.es