Advertisement

Abstract

Let Λ be an invariant set for a C r diffeomorphism f of a manifold M. We say that Λ is a hyperbolic set for f if there is a continuous splitting of the tangent bundle of M restricted to Λ, TM Λ, which is Tf invariant:
$$T{M_\Lambda } = {E^s} \oplus {E^u};\;Tf\left( {{E^s}} \right) = {E^s};\;Tf\left( {{E^u}} \right) = {E^u};$$
and for which there are constants c and λ, c>0 and 0<λ <1, such that
$$\begin{gathered} \left\| {{{\left. {T{f^n}} \right|}_{{E^s}}}} \right\| < c{\lambda ^n},\;n \geqslant 0, \hfill \\ \left\| {{{\left. {T{f^{ - n}}} \right|}_{{E^u}}}} \right\| < c{\lambda ^n},\;n \geqslant 0. \hfill \\ \end{gathered} $$

Keywords

Periodic Orbit Periodic Point Global Stability Finite Sequence Vertical Band 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [4.1]
    Anosov, D. V., Geodesic flows on compact manifolds of negative curvature, Trudy Mat. Inst. Steklov 90 (1967); Proc. Steklov Inst. Math. (transl.) (1969).Google Scholar
  2. [4.2]
    Hirsch, M. W. and Pugh, C. C., Stable manifolds and hyperbolic sets, in Global Analysis,Vol. XIV (Proceedings of Symposia in Pure Mathematics), American Mathematical Society, Providence, R. I., 1970, p.133.Google Scholar
  3. [4.3]
    Levinson, N., A second-order differential equation with singular solutions, Ann. of Math. 50 (1949), 126.MathSciNetCrossRefGoogle Scholar
  4. [4.4]
    Mather, J., Characterization of Anosov diffeomorphisms, Nederl. Akad. Wetensch. Indag. Math. 30 (1968), 479.MathSciNetGoogle Scholar
  5. [4.5]
    Shub, M., Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91 (1969), 175.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [4.6]
    Smale, S., Diffeomorphisms with many periodic points, in Differential and Combinatorial Topology, a Symposium in Honor of M. Morse, S. S. Cairns (Ed.), Princeton University Press, Princeton, N. J., 1965, p. 63.Google Scholar
  7. [4.7]
    Williams, R. F., Expanding attractors, Institut Hautes Études Sei. Publ. Math. 43 (1974), 169.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Michael Shub
    • 1
  1. 1.Thomas J. Watson Research CenterIBMYorktown HeightsUSA

Personalised recommendations