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Shadowing, expansiveness and hyperbolic homeomorphisms

Part of: Connections with other structures, applications

Published online by Cambridge University Press:  09 April 2009

Jerzy Ombach
Jerzy Ombach
Affiliation:
Instytut Matematyki Uniwersytet Jagiellońskiul. Reymonta 4, 30 059 KrakóPoland e-mail: ombach@im.uj.edu.pl
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Abstract

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The purpose of this paper is to complete results concerning the class ℋ of expansive homeomorphisms having the pseudo orbits tracing property on a compact metric space. We show that hyperbolic homeomorphisms introduced by Mañé in [8] are exactly those in the class ℋ then by the result of [12, 20] they form a class equal to the Smale space introduced by Ruelle in [18]. Next, assuming that the phase space is a smooth manifold, we show that a diffeomorphism is Anosov if and only if it is in the class ℋ and is a lower semi-continuity point of the map which assigns to any diffeomorphism the supremum of its expansive constants (possibly zero). Then we discuss the behavior of the dynamical systems generated by homeomorphisms from ℋ near their basic sets.

MSC classification

Secondary: 54H20: Topological dynamics
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 61 , Issue 1 , August 1996 , pp. 57 - 72
Copyright
Copyright © Australian Mathematical Society 1996

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