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

  • Viacheslav Z. GrinesEmail author
  • Timur V. Medvedev
  • Olga V. Pochinka
Chapter
  • 818 Downloads
Part of the Developments in Mathematics book series (DEVM, volume 46)

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, 2, 3, 4, 7, 10, 13, 14, 15, 16, 17, 18, 19, 20].

Keywords

Periodic Point North Pole Hyperbolic Attractor Smale Horseshoe Anosov Diffeomorphism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Viacheslav Z. Grines
    • 1
    Email author
  • Timur V. Medvedev
    • 2
    • 3
  • Olga V. Pochinka
    • 1
  1. 1.Department of Fundamental MathematicsNational Research University Higher School of EconomicsNizhny NovgorodRussia
  2. 2.Department of Differential Equations, Mathematical and Numerical AnalysisNizhny Novgorod State UniversityNizhny NovgorodRussia
  3. 3.Laboratory of Algorithms and Technologies for Networks AnalysisNational Research University Higher School of EconomicsNizhny NovgorodRussia

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