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

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


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].


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.


  1. 1.
    Anosov, D.: Geodesic flows on closed Riemann manifolds with negative curvature, Proc. Steklov Inst. Math., vol. 90. MAIK Nauka/Interperiodica, Pleiades Publishing, Moscow; Springer, Heidelberg (1967)Google Scholar
  2. 2.
    Anosov, D.: About one class of invariant sets of smooth dynamical systems. Proc. Int. Conf. on Non-linear Oscil. 2, 39–45 (1970)Google Scholar
  3. 3.
    Aranson, S., Grines, V.: The topological classification of cascades on closed two-dimensional manifolds. Russ. Math. Surv. 45(1), 1–35 (1990). doi: 10.1070/RM1990v045n01ABEH002322 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Aranson, S., Grines, V.: Dynamical systems with hyperbolic behavior. In: Itogi Nauki Tekhniki; Sovremennye Problemy Matematiki, Fundamental’nye Napravleniya, Dynamical Systems 9, VINITI, Akad. Nauk SSSR, Vol. 66, Moscow, (1991), p.148–187 (Russian), English translation in Encyclopaedia of Mathematical Sciences, Dynamical Systems IX, pp. 141–175. Springer-Verlag-Berlin-Heidelberg (1995)Google Scholar
  5. 5.
    Brin, M., Stuck, G.: Introduction to Dynamical Systems. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  6. 6.
    Farrell, F.T., Jones, L.: Anosov diffeomorphisms constructed from \(\pi _1 Diff(S^n)\). Topology 17(3), 273–282 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Franks, J.: Anosov diffeomorphisms. Proc. Symp. Pure Math. 14, 61–94 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Franks, J., Williams, B.: Anomalous Anosov flows. In: Global theory of dynamical systems, pp. 158–174. Springer (1980)Google Scholar
  9. 9.
    Gibbons, J.C.: One-dimensional basic sets in the three-sphere. Trans. Am. Math. Soc. 164, 163–178 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grines, V.: The topological conjugacy of diffeomorphisms of a two-dimensional manifold on one-dimensional orientable basic sets. I. Tr. Mosk. Mat. O.-va 32, 35–60 (1975)MathSciNetGoogle Scholar
  11. 11.
    Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  12. 12.
    Ku, Y.H., Hyun, J.K., John, B.L.: Eventually periodic points of infra-nil endomorphisms. Fixed Point Theory Appl. 2010 (2010)Google Scholar
  13. 13.
    Manning, A.: There are no new Anosov diffeomorphisms on tori. Am. J. Math. 96, 422–429 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Newhouse, S.E.: On codimension one Anosov diffeomorphisms. Am. J. Math. 92(3), 761–770 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Plykin, R.: The topology of basis sets for Smale diffeomorphisms. Mathematics of the USSR, Sbornik 13, 297–307 (1971). doi: 10.1070/SM1971v013n02ABEH001026 CrossRefzbMATHGoogle Scholar
  16. 16.
    Plykin, R.: Sources and sinks of A-diffeomorphisms of surfaces. Mathematics of the USSR, Sbornik 23, 233–253 (1975). doi: 10.1070/SM1974v023n02ABEH001719 CrossRefzbMATHGoogle Scholar
  17. 17.
    Plykin, R.: Hyperbolic attractors of diffeomorphisms. Russ. Math. Surv. 35(3), 109–121 (1980). doi: 10.1070/RM1980v035n03ABEH001702 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Plykin, R.: On the geometry of hyperbolic attractors of smooth cascades. Russ. Math. Sur. 39(6), 85–131 (1984). doi: 10.1070/RM1984v039n06ABEH003182 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, vol. 28. CRC Press, Boca Raton (1999)zbMATHGoogle Scholar
  20. 20.
    Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73(6), 747–817 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Vietoris, L.: Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen. Math. Ann. 97(1), 454–472 (1927)MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations