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Smale Regular and Chaotic A-Homeomorphisms and A-Diffeomorphisms

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Abstract

We introduce Smale A-homeomorphisms that include regular, semichaotic, chaotic, and superchaotic homeomorphisms of a topological \(n\)-manifold \(M^{n}\), \(n\geqslant 2\). Smale A-homeomorphisms contain axiom A diffeomorphisms (in short, A-diffeomorphisms) provided that \(M^{n}\) admits a smooth structure. Regular A-homeomorphisms contain all Morse – Smale diffeomorphisms, while semichaotic and chaotic A-homeomorphisms contain A-diffeomorphisms with trivial and nontrivial basic sets. Superchaotic A-homeomorphisms contain A-diffeomorphisms whose basic sets are nontrivial. The reason to consider Smale A-homeomorphisms instead of A-diffeomorphisms may be attributed to the fact that it is a good weakening of nonuniform hyperbolicity and pseudo-hyperbolicity, a subject which has already seen an immense number of applications.

We describe invariant sets that determine completely the dynamics of regular, semichaotic, and chaotic Smale A-homeomorphisms. This allows us to get necessary and sufficient conditions of conjugacy for these Smale A-homeomorphisms (in particular, for all Morse – Smale diffeomorphisms). We apply these necessary and sufficient conditions for structurally stable surface diffeomorphisms with an arbitrary number of expanding attractors. We also use these conditions to obtain a complete classification of Morse – Smale diffeomorphisms on projective-like manifolds.

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ACKNOWLEDGMENTS

We thank the unknown reviewers for very useful remarks which improved the text.

Funding

This work was supported by the Laboratory of Dynamical Systems and Applications of the National Research University Higher School of Economics of the Ministry of Science and Higher Education of the RF, grant ag. no. 075-15-2019-1931.

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  1. National Research University Higher School of Economics, ul. Bolshaya Pecherskaya 25/12, 603005, Nizhny Novgorod, Russia

    Vladislav S. Medvedev & Evgeny V. Zhuzhoma

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  1. Vladislav S. Medvedev
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  2. Evgeny V. Zhuzhoma
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Correspondence to Vladislav S. Medvedev or Evgeny V. Zhuzhoma.

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Dedicated to the memory of A. M. Stepin

MSC2010

37D05, 37B35

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Medvedev, V.S., Zhuzhoma, E.V. Smale Regular and Chaotic A-Homeomorphisms and A-Diffeomorphisms. Regul. Chaot. Dyn. 28, 131–147 (2023). https://doi.org/10.1134/S1560354723020016

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