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Global attractor and repeller of Morse-Smale diffeomorphisms

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Abstract

Let f be an orientation-preserving Morse-Smale diffeomorphism of an n-dimensional (n ≥ 3) closed orientable manifold M n. We show the possibility of representing the dynamics of f in a “source-sink” form. The roles of the “source” and “sink” are played by invariant closed sets one of which, A f , is an attractor, and the other, R f , is a repeller. Such a representation reveals new topological invariants that describe the embedding (possibly, wild) of stable and unstable manifolds of saddle periodic points in the ambient manifold. These invariants have allowed us to obtain a classification of substantial classes of Morse-Smale diffeomorphisms on 3-manifolds. In this paper, for any n ≥ 3, we describe the topological structure of the sets A f and R f and of the space of orbits that belong to the set M n \ (A f ∪ R f ).

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Authors and Affiliations

  1. Lobachevsky State University of Nizhni Novgorod, pr. Gagarina 23, Nizhni Novgorod, 603950, Russia

    V. Z. Grines & O. V. Pochinka

  2. Nizhni Novgorod State Pedagogical University, ul. Ul’yanova 1, Nizhni Novgorod, 603950, Russia

    E. V. Zhuzhoma

  3. Research Institute for Applied Mathematics and Cybernetics, Lobachevsky State University of Nizhni Novgorod, ul. Ul’yanova 10, Nizhni Novgorod, 603005, Russia

    V. S. Medvedev

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  1. V. Z. Grines
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  2. E. V. Zhuzhoma
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  4. O. V. Pochinka
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Correspondence to V. Z. Grines.

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Original Russian Text © V.Z. Grines, E.V. Zhuzhoma, V.S. Medvedev, O.V. Pochinka, 2010, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2010, Vol. 271, pp. 111–133.

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Grines, V.Z., Zhuzhoma, E.V., Medvedev, V.S. et al. Global attractor and repeller of Morse-Smale diffeomorphisms. Proc. Steklov Inst. Math. 271, 103–124 (2010). https://doi.org/10.1134/S0081543810040097

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