Chaotic Expansive Homeomorphisms on Closed Orientable Surfaces of Positive Genus
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Chaotic Expansive Homeomorphisms on Closed Orientable Surfaces of Positive Genus Jiehua Mai1 · Song Shao2
Received: 25 March 2015 / Revised: 13 May 2015 / Accepted: 19 May 2015 / Published online: 26 June 2015 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2015
Abstract In this paper we give a new and elementary proof to the following fact: each closed orientable surface of positive genus admits a both chaotic and expansive homeomorphism. Further more, we show that the homeomorphisms given are also weakly mixing. Keywords Expansive homeomorphism · Chaotic homeomorphism · Closed orientable surface Mathematics Subject Classification
37E30 · 54H20 · 37B05
1 Introduction A surface is closed if it is compact, connected and has no boundary. It is well known that any closed surface is homeomorphic either to the sphere, or to the sphere with a finite number of handles added, or to the sphere with a finite number of disks removed and replaced by Möbius strips [5].
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Song Shao [email protected] Jiehua Mai [email protected]
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Institute of Mathematics, Shantou University, Shantou 515063, Guangdong, People’s Republic of China
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Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences and Department of Mathematics, University of Science and Technology of China, Hefei 230026, Anhui, People’s Republic of China
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J. Mai, S. Shao
Definition 1.1 Let f be a homeomorphism of X onto itself, where X is a space with metric d, then f is expansive (with expansive constant c > 0) if to each pair of distinct points x, y, there is an integer n such that d( f n (x), f n (y)) > c. One natural question is what kind of spaces can admit expansive homeomorphisms. It is known that there is no expansive homeomorphism on compact 1-manifold [4,21]. It follows that any compact surface with boundary does not admit an expansive homeomorphism. O’Brien and Reddy [20] showed that each closed orientable surface of positive genus admits an expansive homeomorphism. Hiraide [15] and Lewowicz [16] proved that, on a closed surface, any expansive homeomorphism is conjugate to a pseudo-Anosov map, and hence, there are no expansive homeomorphisms on the sphere, the projective plane, or the Klein bottle. For the higher dimensional manifolds, the problem is more complicated. [23] is one of papers to study expansive homeomorphisms on 3-manifolds. See also [4,17,21] for some results on higher dimensional manifolds. Before giving the definition of a chaotic map, let us recall some basic definitions in dynamics. In this article, natural numbers, integers, real numbers and complex numbers are denoted by N, Z, R and C, respectively. A topological dynamical system is a pair (X, f ), where X is a metric space with a metric d and f is a continuous map from X to itself. The orbit O(x, f ) of a point x under the homeomorphism f : X → X is the set { f n (x) : n ∈ Z}. Similarly, the set O+ (x, f ) = { f n (x) : n ∈ N ∪ {0}} is called the positive semi-orbit of x, and O(x,
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