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Pointwise Topological Stability

Part of: Topological dynamics Smooth dynamical systems: general theory

Published online by Cambridge University Press:  15 August 2018

Namjip Koo ,
Keonhee Lee  and
C. A. Morales
Namjip Koo
Affiliation:
Department of Mathematics, Chungnam National University, Daejeon, 305-764, Republic of Korea (njkoo@cnu.ac.kr; khlee@cnu.ac.kr)
Keonhee Lee
Affiliation:
Department of Mathematics, Chungnam National University, Daejeon, 305-764, Republic of Korea (njkoo@cnu.ac.kr; khlee@cnu.ac.kr)
C. A. Morales*
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, PO Box 68530, 21945-970, Brazil (morales@impa.br)
*
*Corresponding author.
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Abstract

We decompose the topological stability (in the sense of P. Walters) into the corresponding notion for points. Indeed, we define a topologically stable point of a homeomorphism f as a point x such that for any C0-perturbation g of f there is a continuous semiconjugation defined on the g-orbit closure of x which tends to the identity as g tends to f. We obtain some properties of the topologically stable points, including preservation under conjugacy, vanishing for minimal homeomorphisms on compact manifolds, the fact that topologically stable chain recurrent points belong to the periodic point closure, and that the chain recurrent set coincides with the closure of the periodic points when all points are topologically stable. Next, we show that the topologically stable points of an expansive homeomorphism of a compact manifold are precisely the shadowable ones. Moreover, an expansive homeomorphism of a compact manifold is topologically stable if and only if every point is topologically stable. Afterwards, we prove that a pointwise recurrent homeomorphism of a compact manifold has no topologically stable points. Finally, we prove that every chain transitive homeomorphism with a topologically stable point of a compact manifold has the pseudo-orbit tracing property. Therefore, a chain transitive expansive homeomorphism of a compact manifold is topologically stable if and only if it has a topologically stable point.

Keywords

topologically stable pointshadowable pointhomeomorphismmetric space

MSC classification

Primary: 37C50: Approximate trajectories (pseudotrajectories, shadowing, etc.)
Secondary: 37B45: Continua theory in dynamics
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 4 , November 2018 , pp. 1179 - 1191
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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