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
$$\begin{gathered} \left\| {{{\left. {T{f^n}} \right|}_{{E^s}}}} \right\| < c{\lambda ^n},\;n \geqslant 0, \hfill \\ \left\| {{{\left. {T{f^{ - n}}} \right|}_{{E^u}}}} \right\| < c{\lambda ^n},\;n \geqslant 0. \hfill \\ \end{gathered} $$


Periodic Orbit Periodic Point Global Stability Finite Sequence Vertical Band 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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.MathSciNetMATHCrossRefGoogle 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