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The Properties of Nontrivial Basic Sets of A-Diffeomorphisms Related to Type and Dimension

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Dynamical Systems on 2- and 3-Manifolds

Part of the book series: Developments in Mathematics ((DEVM,volume 46))

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Abstract

In this chapter we study orientation preserving A-diffeomorphisms on an orientable compact manifold \(M^n\) (possibly with boundary) with a nontrivial basic set \(\varLambda \) in the interior of \(M^n\). We state some important properties of the basic sets in relation to their type and dimension. These properties are used for the topological classification of the basic sets (including expanding attractors and contracting repellers) as well as for important classes of structurally stable diffeomorphisms. We present the constructions of classical A-diffeomorphisms with basic sets of codimension one: the DA-diffeomorphism, the diffeomorphism with the Plykin attractor, the diffeomorphism with the Smale “horseshoe”, the diffeomorphism with the Smale-Williams solenoid. The results of this chapter are based on [1–4, 7, 10, 13–20].

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Notes

  1. 1.

    There is a hypothesis that the periodic trajectories of every Anosov diffeomorphism are dense in the ambient manifold and therefore the ambient manifold is its only basic set. From [6, 7, 12–14] it follows that this is true for all known Anosov diffeomorphisms. But J. Franks and R. Williams in [8] constructed a class of Anosov flows for which the non-wandering set does not coincide with the ambient manifold.

  2. 2.

    The given descriptions of these diffeomorphisms are not complete. Detailed explanations could be found, for example, in [5, 11, 19, 20].

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Authors and Affiliations

  1. Department of Fundamental Mathematics, National Research University Higher School of Economics, Nizhny Novgorod, Russia

    Viacheslav Z. Grines & Olga V. Pochinka

  2. Department of Differential Equations, Mathematical and Numerical Analysis, Nizhny Novgorod State University, Nizhny Novgorod, Russia

    Timur V. Medvedev

  3. Laboratory of Algorithms and Technologies for Networks Analysis, National Research University Higher School of Economics, Nizhny Novgorod, Russia

    Timur V. Medvedev

Authors
  1. Viacheslav Z. Grines
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  2. Timur V. Medvedev
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  3. Olga V. Pochinka
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Correspondence to Viacheslav Z. Grines .

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Grines, V.Z., Medvedev, T.V., Pochinka, O.V. (2016). The Properties of Nontrivial Basic Sets of A-Diffeomorphisms Related to Type and Dimension. In: Dynamical Systems on 2- and 3-Manifolds. Developments in Mathematics, vol 46. Springer, Cham. https://doi.org/10.1007/978-3-319-44847-3_8

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