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Some properties of positive entropy maps

Published online by Cambridge University Press:  15 January 2013

A. ARBIETO  and
C. A. MORALES
A. ARBIETO
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, PO Box 68530, 21945-970, Rio de Janeiro, Brazil (email: arbieto@im.ufrj.br, morales@impa.br)
C. A. MORALES
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, PO Box 68530, 21945-970, Rio de Janeiro, Brazil (email: arbieto@im.ufrj.br, morales@impa.br)
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Abstract

We prove that the stable classes for continuous maps on compact metric spaces have measure zero with respect to any ergodic invariant measure with positive entropy. Then, every continuous map with positive topological entropy on a compact metric space has uncountably many stable classes. We also prove that every continuous map with positive topological entropy of a compact metric space cannot be Lyapunov stable on its recurrent set. For homeomorphisms on compact metric spaces we prove that the sets of heteroclinic points, and sinks in the canonical coordinates case, have zero measure with respect to any ergodic invariant measure with positive entropy. These results generalize those of Fedorenko and Smital [Maps of the interval Ljapunov stable on the set of nonwandering points. Acta Math. Univ. Comenian. (N.S.)60 (1) (1991), 11–14], Huang and Ye [Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos. Topology Appl.117 (3) (2002), 259–272], Reddy [The existence of expansive homeomorphisms on manifolds. Duke Math. J.32 (1965), 627–632], Reddy and Robertson [Sources, sinks and saddles for expansive homeomorphisms with canonical coordinates. Rocky Mountain J. Math.17 (4) (1987), 673–681], Sindelarova [A counterexample to a statement concerning Lyapunov stability. Acta Math. Univ. Comenian. 70 (2001), 265–268], and Zhou [Some equivalent conditions for self-mappings of a circle. Chinese Ann. Math. Ser. A12(suppl.) (1991), 22–27].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 3 , June 2014 , pp. 765 - 776
Copyright
©2013 Cambridge University Press 

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