() @ Applied General Topology c© Universidad Politécnica de Valencia Volume 12, no. 2, 2011 pp. 95-100 Some remarks on chaos in topological dynamics Huoyun Wang ∗ and Heman Fu Abstract Bau-Sen Du introduced a notion of chaos which is stronger than Li- Yorke sensitivity. A TDS (X, f) is called chaotic if there is a positive ε such that for any x and any nonempty open set V of X there is a point y in V such that the pair (x, y) is proximal but not ε-asymptotic. In this article, we show that a TDS (T, f) is transitive but not mixing if and only if (T, f) is Li-Yorke sensitive but not chaotic, where T is a tree. Moreover, we compare such chaos with other notions of chaos. 2010 MSC: 37B05, 54H20, 37B20, 58K15. Keywords: sensitivity, chaos, tree maps. 1. Introduction Throughout this paper a topological dynamical system is a pair (X, f) (TDS for short), where X is a compact metric space with a metric d and f : X → X is a continuous surjective map. A TDS (X, f) is nontrivial if X contains at least two points. Chaotic behavior is a manifestation of the complexity of the dynamical system. Now we recall some concepts of complexity. A TDS (X, f) is sensitive [3], if there exists a positive ε such that for any x in X and any open neighborhood U of x, there exist y ∈ U and a positive integer n with d(fn(x), fn(y)) > ε. Let ε be a positive number. A subset C in X is a Li-Yorke ε-scrambled set of a TDS (X, f), if any pair (x, y) of distinct points x and y in C is proximal but not ε-asymptotic, that is, lim inf n→∞ d(fn(x), fn(y)) = 0 and lim sup n→∞ d(fn(x), fn(y)) > ε. ∗Supported by National Nature Science Funds of China (11071084), and Guangzhou Ed- ucation Bureau (08C016). 96 H. Wang and H. Fu A TDS (X, f) is Li-Yorke ε-chaotic [5], if it has an uncountable Li-Yorke ε-scrambled set. In 2003, Akin and Kolyada [1] introduced the notion of Li-Yorke sensitivity which links the Li-Yorke version of chaos with sensitivity. A TDS (X, f) is called Li-Yorke sensitive [1] if there is a positive ε such that every x in X is a limit of points y in X such that the pair (x, y) is proximal but not ε-asymptotic. What is the nature of chaos? Various people have various understandings. In [4], Bau-Sen Du believed that chaos should involve not only nearby points could diverge apart but also faraway points could get close to each other. Therefore in 2006, he [4] proposed a new definition of chaos as follows, which is stronger than Li-Yorke sensitivity. A TDS (X, f) is called chaotic [4] if there is a positive ε such that for any x and any nonempty open set V of X there is a point y in V such that the pair (x, y) is proximal but not ε-asymptotic. There is a TDS (X, f) which is Li-Yorke sensitive but not chaotic (see [4, Theorem 4]). The present article goes on studying the nature of chaos, and is written on basis of the preprint [4]. This article is organized as follows. In Section 2, we investigate the chaos of transitive maps on trees. We show that a TDS (T, f) is transitive but not mixing if and only if (T, f) is Li-Yorke sensitive but not chaotic, where T is a tree. Finally, we compare the chaos with other notion of the same. 2. The chaos of transitive maps on trees In this section, the chaos of transitive maps on trees are investigated. By a tree we mean a connected compact one-dimensional polyhedron, which does not contain any subset homeomorphic to a circle and which contains a subset homeomorphic to an interval. Let T be a tree. Given point x ∈ T , we define the valence of x, val(x), as the number of connected components of T − {x}. Each point of valence 1 is an endpoint of T . A subtree of a tree T is a subset of T , which is a tree itself. A TDS (X, f) is (topologically) transitive if for any two nonempty open sets U and V there exists a positive integer n such that fn(U)∩V 6= ∅. A TDS (X, f) is (topologically) weakly mixing if (X × X, f × f) is (topologically) transitive. A TDS (X, f) is (topologically) totally transitive if (X, fn) is transitive for any positive integer n. A TDS (X, f) is (topologically) mixing if for any two nonempty open sets U and V there exists a positive integer N such that fn(U) ∩ V 6= ∅ for any n ≥ N. A TDS (X, f) is minimal if the set orb(x, f) = {fn(x) : n = 0, 1, 2, · · ·} is dense in X for any x of X. The main aim of this section is to prove the following result. Some remarks on chaos in topological dynamics 97 Theorem 2.1. Let T be a tree and let (T, f) be transitive. Then the following results hold. (1) (T, f) is mixing if and only if it is chaotic; (2) (T, f) is not mixing if and only if it is Li-Yorke sensitive but not chaotic. Remark 2.2. Theorem 2.1 may not be true for a general TDS. Let g be an irrational rotation of the unit circle S1, then (S1, g) is minimal and totally transitive but not Li-Yorke sensitive. We need the following lemmas which come from [8] and [2] respectively. Lemma 2.3. Let T be a tree and let (T, f) be transitive. Then P(f) = T , where P(f) denotes the closure of the set of all periodic points of f. Lemma 2.4. Let T be a tree and let (T, f) be transitive. Then exactly one of the following alternatives holds. (1) (T, f) is totally transitive. (2) There is a positive integer n0 such that there are an interior fixed point y and subtrees T1, T2, · · · , Tn0 of T with ∪ n0 i=1Ti = T , Ti ∩ Tj = {y} for i 6= j and f(Ti) = Ti+1(mod n0) for 1 ≤ i ≤ n0. Moreover, (Ti, f n0|Ti) is transitive, 1 ≤ i ≤ n0. Proposition 2.5. Let f : T → T be a tree map. (T, f) is mixing if and only if it is totally transitive. Proof. Let us denote by E(T ) the set of endpoints of the tree T and suppose (T, f) is totally transitive. Suppose that U and V are nonempty open connected subsets of T . We may assume that U is contained in T − E(T ). Since (T, f) is transitive, then P(f) = T by Lemma 2.3. Let y be any periodic point in U which orbit Orb(y, f) is contained in T − E(T ). Let x be any periodic point in V . Let m be a common multiple of the periods of x and y, and set g = fm. Then every point of Orb(y, f) ∪ {x} is a fixed point of g. Let K = ∪∞n=0g n(V ). Then K is a connected subset of T , since x is a fixed point of g. Since (T, g) is transitive, then K is a dense connected subset of T . This implies K contains T − E(T ). For any u ∈ Orb(y, f), there is an integer ku ≥ 0 such that u ∈ g ku(V ). Let k = max{ku : u ∈ Orb(y, f)}. Since every point u of Orb(y, f) is a fixed point of g, then fkm(V ) = gk(V ) which contains Orb(y, f). Thus fn(V ) contains point y for any n ≥ km. This implies fn(V ) ∩ U 6= ∅, hence (T, f) is mixing. Conversely, it is obvious. � Proposition 2.6. Let T be a tree and let (T, f) be transitive. Then exactly one of the following alternatives holds. (1) (T, f) is mixing. (2) There is a positive integer n0 such that there are an interior fixed point y and subtrees T1, T2, · · · , Tn0 of T with ∪ n0 i=1Ti = T , Ti ∩ Tj = {y} for 98 H. Wang and H. Fu i 6= j and f(Ti) = Ti+1(mod n0) for 1 ≤ i ≤ n0. Moreover, (Ti, f n0|Ti) is mixing, 1 ≤ i ≤ n0. Proof. Suppose that (T, f) is transitive. If (T, f) is totally transitive, then (T, f) is mixing by Proposition 2.5. Otherwise, there is a positive integer n0 such that there are an interior fixed point y and subtrees T1, T2, · · · , Tn0 of T with ∪n0i=1Ti = T , Ti ∩ Tj = {y} for i 6= j and f(Ti) = Ti+1(mod n0) for 1 ≤ i ≤ n0. Moreover, (Ti, f n0|Ti) is transitive, 1 ≤ i ≤ n0. Then f n0|Ti satisfies the condition (1) of Lemma 2.4, since y is a fixed point of fn0|Ti, and y is the endpoint of Ti. Hence (Ti, f n0|Ti) is mixing by Lemma 2.4 and Proposition 2.5. � By [1, Theorem 3.4 and Lemma 3.8], the following lemma holds. Lemma 2.7. If a nontrivial TDS (X, f) is weakly mixing, then it is chaotic. Proof of Theorem 2.1. If (T, f) is mixing, then (T, f) is chaotic by Lemma 2.7. Otherwise, by Proposition 2.6 there are a positive integer n0 such that there is an interior fixed point y and subtrees T1, T2, · · · , Tn0 of T with ∪ n0 i=1Ti = T , Ti ∩ Tj = {y} for i 6= j and f(Ti) = Ti+1(mod n0) for 1 ≤ i ≤ n0. Moreover, (Ti, f n0|Ti) is mixing, 1 ≤ i ≤ n0. Since every (Ti, f n0|Ti) is chaotic, hence (T, f) is Li-Yorke sensitive. Next we show that (T, f) is not chaotic. Let x be a periodic point of f in the interior of T1. Let V be an open subset of T which is contained in T2. Since x and V are jumping alternatively and never get close to each other, then (T, f) is not chaotic. Hence, Theorem 2.1 holds. 2 3. Comparison of various notions of chaos In this section X will denote a general compact metric space. Below, we discuss the interrelations between the notions of chaos. A TDS (X, f) is Devaney’s chaotic [3] if it is transitive and the set of periodic points of f is dense in X. Recall that a Mycielski set is a countable union of Cantor sets, while a Cantor set is a set homeomorphic to the standard middle-third Cantor set on the real line. The following lemmas come from [1] and [6] respectively. Lemma 3.1. For a TDS (X, f) the following conditions are equivalent. 1. (X, f) is sensitive. 2. There exists a positive ε such that {(x, y) ∈ X×X : lim supn→∞ d(f n(x), fn(y)) > ε} is a dense Gδ set of X × X. Lemma 3.2 (Mycielski). Let X be a separable complete metric space without isolated point. If for each natural number n, Rn is a residual set of X n, then there is a Mycielski set K of X such that (1) for any nonempty open set U of X, K ∩ U contains a nonempty perfect set; (2) for each natural number n, for all x1, x2, · · · , xn mutually distinct points in K, (x1, x2, · · · , xn) ∈ Rn. Some remarks on chaos in topological dynamics 99 Theorem 3.3. If a TDS (X, f) is chaotic, then there is a dense Mycielski set K in X such that K is a Li-Yorke ε-scrambled set for some positive ε. Hence, (X, f) is Li-Yorke ε-chaotic. Proof. Since (X, f) is chaotic, then (X, f) is sensitive, this implies X without isolated point. There is a positive ε such that C1(ε) := {(x, y) ∈ X × X : lim supn→∞ d(f n(x), fn(y)) > ε} is a dense Gδ set of X × X, by Lemma 3.1. Moreover, C2 := {(x, y) ∈ X × X : lim infn→∞ d(f n(x), fn(y)) = 0} is a dense Gδ set of X × X because (X, f) is chaotic. Put R2 = C1(ε) ∩ C2. So Theorem 3.3 holds by Lemma 3.2. � Remark 3.4. There is a TDS with positive topological entropy, which is Li- Yorke ε-chaotic and Devaney’s chaotic but not chaotic. Let f(x) = 1 2 + 2x if 0 ≤ x ≤ 1 4 ; f(x) = 3 2 − 2x if 1 4 ≤ x ≤ 1 2 ; f(x) = 1 − x if 1 2 ≤ x ≤ 1. Then f : [0, 1] → [0, 1] is transitive but not mixing. Since f is transitive, then the set of periodic points of f is dense in X. This implies that ([0, 1], f) is Devaney’s chaotic. Because f has positive topological entropy, then ([0, 1], f) is Li-Yorke ε-chaotic for some positive ε but not chaotic. Remark 3.5. There is a TDS which is chaotic with zero topological entropy. A minimal system (X, f) which is weakly mixing and has zero topological entropy has been built, such as in [7]. Then the minimal system (X, f) is chaotic, and has zero topological entropy. Remark 3.6. If a TDS (X, f) is Li-Yorke sensitive, the product system (X × Y, f × g) is Li-Yorke sensitive for any TDS (Y, g) (see [1, Theorem 3.11]). However, for any TDS (X, f) which is chaotic, there is a TDS (Y, g) such that the product system (X × Y, f × g) is not chaotic. Actually, let g be an irrational rotation of the unit circle S1. The product system (X × S1, f × g) is Li-Yorke sensitive but not chaotic for any chaotic TDS (X, f). Remark 3.7. There is a TDS (X, σ) which is chaotic but not Devaney’s chaotic. Let ( ∑ 2, σ) be the full shift over two letters {0, 1}. Let A1 = 1, A2 = 101, · · · , An+1 = An0 nAn. Then x = limn→∞ AnAn · · · ∈ ∑ 2. Put X = Orb(x, σ). Then (X, σ) is mixing and has a fixed point 00 · · · which is the unique minimal subset of X (see [9]). (X, σ) is chaotic but not Devaney’s chaotic since the set of periodic points of σ is not dense in X. There exists a TDS which is chaotic but not transitive, a fortiori not weakly mixing(see [4, Theorem 5]). However we have Remark 3.8. If a TDS (X, f) is minimal, then (X, f) is chaotic if and only if it is weakly mixing. If (X, f) is chaotic, then the proximal cell P(f)(x) is dense in X for any x of X, where P(f)(x) = {y ∈ X : lim infn→∞ d(f n(x), fn(y)) = 0}. Hence (X, f) is weakly mixing by [1, Theorem 3.7]. Conversely, it is clear. 100 H. Wang and H. Fu Below, we summarize the interrelations between the notions of chaos, see Figure 1. positive entropy ; : chaos ; : Devaney’s chaos weakly mixing ⇓ ⇑+minimal w 6t u 6v Li-Yorke sensitivity Li-Yorke ε-chaos Figure 1. Relations of various notions of chaos Acknowledgements. The authors would like to thank the referee for the careful reading and many valuable comments. References [1] E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity 16 (2003), 1421–1433. [2] L. Alseda, S. Kolyada, J. Llibre and L. Snoha, Entropy and Periodic points for tree maps, Trans. Amer. Math. Soc. 351 (1997), 1551–1573. [3] R. Devaney, Chaotic Dynamical Systems, Addison-Wesley, Redwood City, 1980. [4] B. Du, On the nature of chaos, arXiv: math.DS/0602585 v1 26 Feb 2006. [5] T. Li and J. Yorke, Period 3 implies chaos, Amer. Math. Monthly 82 (1975), 985–992. [6] J. Mycielski, Independent sets in topological algebras, Fund. Math. 55 (1964), 139–147. [7] L. Wang, Z. Chen and G. Liao, The complexity of a minimal sub-shift on symbolic spaces, J. Math. Anal. Appl. 37 (2006), 136–145. [8] X. Ye, The center and the depth of the center of a tree map, Bull. Austral. Math. Soc. 48 (1993), 347–350. [9] X. Ye, W. Huang and S. Shao, An Introduction to Topolgical Dynamics, Science Press, Bejing, 2008. [Chinese] (Received September 2010 – Accepted July 2011) Huoyun Wang (wanghuoyun@126.com) Department of Mathematics, Guangzhou University, Guangzhou, 510006, Peo- ple’s Republic of China Heman Fu (dbfhm@163.com) College of Mathematics and Information Sciences, Zhaoqing University, Zhao- qing, 526061, People’s Republic of China Some remarks on chaos in topological dynamics. By H. Wang and H. Fu