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Persistent properties from the Gromov–Hausdorff viewpoint

Part of: Topological dynamics

Published online by Cambridge University Press:  02 December 2024

Abdul Gaffar Khan ,
Carlos Arnoldo Morales Open the ORCID record for Carlos Arnoldo Morales [Opens in a new window]  and
Tarun Das
Abdul Gaffar Khan
Affiliation:
Kirori Mal College, University of Delhi, Delhi, India
Carlos Arnoldo Morales
Affiliation:
Beijing Advanced Innovation Center for Future Blockchain and Privacy Computing, Beihang University, Beijing, China
Tarun Das*
Affiliation:
Department of Mathematics, University of Delhi, Delhi, India
*
Corresponding author: Tarun Das, email: tarukd@gmail.com
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Abstract

In this paper, we introduce topologically IGH-stable, IGH-persistent,average IGH-persistent and pointwise weakly topologically IGH-stable homeomorphisms of compact metric spaces. We prove that every topologically IGH-stable homeomorphism is topologically stable and every expansive topologically stable homeomorphism of a compact manifold is topologically IGH-stable. We further prove that every equicontinuous pointwise weakly topologically IGH-stable homeomorphism is IGH-persistent and every pointwise minimally expansive IGH-persistent homeomorphism is pointwise weakly topologically IGH-stable. Finally, we prove that every mean equicontinuous pointwise weakly topologically IGH-stable homeomorphism is average IGH-persistent.

Keywords

Gromov–Hausdorffpersistenttopologically stable

MSC classification

Primary: 37B25: Lyapunov functions and stability; attractors, repellers
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 4 , November 2024 , pp. 1229 - 1240
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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References

Aoki, N. and Hiraide, K., Topological theory of dynamical systems: recent advances, North-Holland Mathematical Library, Volume 52, (North-Holland Publishing Co, Amsterdam, 1994).Google Scholar
Arbieto, A. and Rojas, C. A., Topological stability from Gromov-Hausdorff viewpoint, Discrete Contin. Dyn. Syst. 37 (7) (2017), 3531–3544.CrossRefGoogle Scholar
Carvalho, B. and Cordeiro, W., N-Expansive homeomorphisms with the shadowing property, J. Differential Equations 261 (6) (2016), 3734–3755.CrossRefGoogle Scholar
Dong, M., Lee, K. and Morales, C., Pointwise topological stability and persistence, J. Math. Anal. Appl. 480 (2) (2019), 1–12.CrossRefGoogle Scholar
Gromov, M., Metric structures for Riemannian and non-Riemannian spaces, Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, Volume 152, (Birkhauser Boston, Inc, Boston, MA, 1999).Google Scholar
Khan, A. G. and Das, T., Average shadowing and persistence in pointwise dynamics, Topology Appl. 292 (2021), 1–13.CrossRefGoogle Scholar
Khan, A. G. and Das, T., Persistence and expansivity through pointwise dynamics, Dyn. Syst. 36 (1) (2021), 79–87.CrossRefGoogle Scholar
Khan, A. G. and Das, T., Stability theorems in pointwise dynamics, Topology Appl. 320 (2022), .CrossRefGoogle Scholar
Koo, N., Lee, K. and Morales, C. A., Pointwise topological stability, Proc. Edinb. Math. Soc. (2) 61 (4) (2018), 1179–1191.CrossRefGoogle Scholar
Lee, J. and Rojas, A., A topological shadowing theorem, Proc. Indian Acad. Sci. Math. Sci. 132(1) (2022), 1–9.CrossRefGoogle Scholar
Lewowicz, J., Persistence in expansive systems, Ergodic Theory Dynam. Systems 3 (4) (1983), 567–578.CrossRefGoogle Scholar
Li, J., Tu, S. and Ye, X., Mean equicontinuity and mean sensitivity, Ergodic Theory Dynam. Systems 35 (8) (2015), 2587–2612.CrossRefGoogle Scholar
Morales, C. A., Shadowable points, Dyn. Syst. 31 (3) (2016), 347–356.CrossRefGoogle Scholar
Utz, W. R., Unstable homeomorphisms, Proc. Amer. Math. Soc. 1 (6) (1950), 769–774.CrossRefGoogle Scholar
Walters, P., On the pseudo orbit tracing property and its relationship to stability. In the structure of attractors in dynamical systems, , (Springer, Berlin, Heidelberg, 1978).CrossRefGoogle Scholar