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Local Topological Stability for Diffeomorphisms

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Abstract

Let \(f:M\rightarrow M\) be a diffeomorphism of compact smooth Riemannian manifold M,  an let \(\Lambda \subset M\) be a closed f-invariant set. We obtain conditions for \(\Lambda \) to be topologically stable which is called \(\Lambda \)-topologically stable. Moreover, we prove that if f is \(C^1\) robustly \(\Lambda \)-topologically stable then \(\Lambda \) satisfies star condition for f. Then in the above, if a closed f-invariant set \(\Lambda \) is chain transitive (or transitive) then it is hyperbolic for f.

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Acknowledgements

The author would like thanks to C. A. Morales for his valuable comments and suggestions. Also, the author thanks both referees for very useful and valuable remarks. This work is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (No. 2017R1A2B4001892 and 2020R1F1A1A01051370).

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  1. Department of Marketing Big Data and Mathematics, Mokwon University, Daejeon, 302-729, Korea

    Manseob Lee

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Lee, M. Local Topological Stability for Diffeomorphisms. Qual. Theory Dyn. Syst. 22, 51 (2023). https://doi.org/10.1007/s12346-023-00755-6

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