Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-23T17:32:07.776Z Has data issue: false hasContentIssue false
  • English
  • Français

Pointwise Topological Stability

Part of: Smooth dynamical systems: general theory Topological dynamics

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Aoki, N. and Hiraide, K., Topological theory of dynamical systems. Recent advances. North-Holland Mathematical Library, Volume 52 (North-Holland, Amsterdam, 1994).Google Scholar
2Aponte, J., Shadowable, topologically stable and distal points for flows. Thesis, Universidade Federal do Rio de Janeiro UFRJ (2017).Google Scholar
3Auslander, J., Minimal flows and their extensions. North-Holland Mathematics Studies, Volume 153, Notas de Matemática [Mathematical Notes] Volume 122 (North-Holland, Amsterdam, 1988).Google Scholar
4Choi, S. K., Chu, C. and Lee, K., Recurrence in persistent dynamical systems, Bull. Aust. Math. Soc. 43(3) (1991), 509–517.Google Scholar
5Ferraz, E., Uniform limits and pointwise dynamics. MSc Dissertation, Universidade Federal do Rio de Janeiro UFRJ (2017).Google Scholar
6Kagwaguchi, N., Quantitative shadowable points, Dyn. Syst. 32(4) ( 2017), 504–518.Google Scholar
7Lee, K. and Park, J. S., Points which satisfy the closing lemma, Far East J. Math. Sci. 3(2) (1995), 171–177.Google Scholar
8Lewowicz, J., Dinámica de los homeomorfismos expansivos (in Spanish) [Dynamics of expansive homeomorphisms] Monografías del Instituto de Matemática y Ciencias Afines [Monographs of the Institute of Mathematics and Related Sciences] Volume 36 (Instituto de Matemática y Ciencias Afines, IMCA, Lima; Pontificia Universidad Católica del Perú, Lima, 2003).Google Scholar
9Morales, C. A., Shadowable points, Dyn. Syst. 31(3) (2016), 347–356.Google Scholar
10Pilyugin, S. Y., Shadowing in dynamical systems, Lecture Notes in Mathematics, Volume 1706 (Springer-Verlag, Berlin, 1999).Google Scholar
11Reddy, W. L., Pointwise expansive homeomorphisms, J. Lond. Math. Soc. (2) 2 (1970), 232–236.Google Scholar
12Utz, W. R., Unstable homeomorphisms, Proc. Amer. Math. Soc. 1 (1950), 769–774.Google Scholar
13Walters, P., On the pseudo-orbit tracing property and its relationship to stability, In The structure of attractors in dynamical systems (Proc. Conf., North Dakota State University, Fargo, ND, 1977), Lecture Notes in Mathematics, Volume 668, pp. 231–244 (Springer, Berlin, 1978).Google Scholar
14Yano, K., Topologically stable homeomorphisms of the circle, Nagoya Math. J. 79 (1980), 145–149.Google Scholar
15Ye, X. and Zhang, G., Entropy points and applications, Trans. Amer. Math. Soc. 359(12) (2007), 6167–6186.Google Scholar