×

Pointwise persistence and shadowing. (English) Zbl 1447.37027

The authors introduce and study the notion of persistent shadowing property for homeomorphisms on compact metric spaces. More precisely, a homeomorphism \(f\) has the persistent shadowing property if for any \(\varepsilon >0\), there is \(\delta >0\) such that any \(\delta\)-pseudo orbit of a homeomorphism \(g\) with \(d_{C^0}(f,g)\leq \delta\) can be \((g,\varepsilon)\)-shadowed. They characterize the notion of persistent shadowing property from a pointwise viewpoint. It is proved that every pointwise persistent homeomorphism with shadowing property is persistent. Moreover, they show that any expansive homeomorphism with the persistent shadowing property is \(s\)-topologically stable.

MSC:

37B65 Approximate trajectories, pseudotrajectories, shadowing and related notions for topological dynamical systems
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B02 Dynamics in general topological spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aoki, N., The splitting of zero-dimensional automorphisms and its application, Colloq. Math., 49, 2, 161-173 (1985) · Zbl 0592.22008 · doi:10.4064/cm-49-2-161-173
[2] Bowen, R., \( \omega \)-limit sets for axiom a diffeomorphisms, J. Differ. Equ., 18, 2, 333-339 (1975) · Zbl 0315.58019 · doi:10.1016/0022-0396(75)90065-0
[3] Dong, M.; Jung, W.; Morales, C., Eventually shadowable points, Qual. Theory Dyn. Syst., 19, 16 (2020) · Zbl 1432.37039 · doi:10.1007/s12346-020-00367-4
[4] Dong, M.; Lee, K.; Morales, CA, Pointwise topological stability and persistence, J. Math. Anal. Appl., 480, 2, 123334 (2019) · Zbl 1425.37010 · doi:10.1016/j.jmaa.2019.07.024
[5] Good, C.; Meddaugh, J., Orbital shadowing, internal chain transitivity and \(\omega \)-limit sets, Ergod. Theory Dyn. Syst., 38, 1, 143-154 (2018) · Zbl 1387.37012 · doi:10.1017/etds.2016.30
[6] Gu, R., Uniform pseudo-orbit tracing property and orbit stability, Ann. Differ. Equ., 13, 4, 350-359 (1997) · Zbl 0906.58021
[7] Kawaguchi, N., On the topological stability and shadowing in zero-dimensional spaces, Discrete Contin. Dyn. Syst., 39, 5, 2743-2761 (2019) · Zbl 1472.37019 · doi:10.3934/dcds.2019115
[8] Kulczycki, M., Uniform pseudo-orbit tracing property for homeomorphisms and continuous mappings, Ann. Polon. Math., 75, 1, 1-6 (2000) · Zbl 0962.37009 · doi:10.4064/ap-75-1-1-6
[9] Lewowicz, J., Persistence in expansive systems, Ergod. Theory Dyn. Syst., 3, 4, 567-578 (1983) · Zbl 0529.58021 · doi:10.1017/S0143385700002157
[10] Morales, CA, Shadowable points, Dyn. Syst., 31, 3, 347-356 (2016) · Zbl 1369.37030 · doi:10.1080/14689367.2015.1131813
[11] Pilyugin, SY; Rodionova, AA; Sakai, K., Orbital and weak shadowing properties, Discrete Contin. Dyn. Syst., 9, 2, 287-308 (2003) · Zbl 1015.37020 · doi:10.3934/dcds.2003.9.287
[12] Robinson, C., Stability theorems and hyperbolicity in dynamical systems, Proceedings of the Regional Conference on the Application of Topological Methods in Differential Equations (Boulder, Colo., 1976), Rocky Mt. J. Math., 7, 3, 425-437 (1977) · Zbl 0375.58016 · doi:10.1216/RMJ-1977-7-3-425
[13] Sakai, K., The \(C^1\) uniform pseudo-orbit tracing property, Tokyo J. Math., 15, 1, 99-109 (1992) · Zbl 0763.58022 · doi:10.3836/tjm/1270130253
[14] Sakai, K., Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math., 31, 2, 373-386 (1994) · Zbl 0820.58045
[15] Sakai, K., Diffeomorphisms with persistency, Proc. Am. Math. Soc., 124, 7, 2249-2254 (1996) · Zbl 0861.54035 · doi:10.1090/S0002-9939-96-03275-3
[16] Sakai, K., Kobayashi, H.: On persistent homeomorphisms, Dynamical systems and nonlinear oscillations (Kyoto, 1985), 1-12, World Scientific Advanced Series in Dynamical Systems, 1. World Sci. Publishing, Singapore (1986)
[17] Utz, WR, Unstable homeomorphisms, Proc. Am. Math. Soc., 1, 769-774 (1950) · Zbl 0040.09903 · doi:10.1090/S0002-9939-1950-0038022-3
[18] Walters, 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 Univ., Fargo, N.D., 1977). Lecture Notes in Mathematics, pp. 231-244, vol. 668. Springer, Berlin (1978) · Zbl 0403.58019
[19] Yano, K., Topologically stable homeomorphisms of the circle, Nagoya Math. J., 79, 145-149 (1980) · Zbl 0417.58008 · doi:10.1017/S0027763000018997
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.