One way to understand the recurrence of a homeomorphism is to try to build up the underlying manifold from simpler pieces, each of which isolates and asymptotically specifies an invariant set. With this in mind we make several definitions. Given a homeomorphism f, of M, a filtration M adapted to f is a nested sequence Ø = M 0M 1 ⊂ ··· ⊂ M k = M of smooth, compact codimension 0 submanifolds with boundary of M, such that f(M i ), ⊂ Int M i .


Limit Point Disjoint Union Global Stability Compact Neighborhood Nest Sequence 
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  1. [2.1]
    Newhouse, S., On hyperbolic limit sets, Trans. Amer. Math. Soc. 167 (1972), 125.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2.2]
    Nitecki, Z. and Shub, M., Filtrations, decompositions, and explosions, Amer. J. Math. 107 (1975), 1029.Google Scholar
  3. [2.3]
    Palis, J., On Morse—Smale dynamical systems, Topology 8 (1969), 385.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [2.4]
    Rosenberg, H., A generalization of the Morse—Smale inequalities, Bull Amer. Math. Soc. 70 (1964), 422.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [2.5]
    Shub, M., Stability and genericity for diffeomorphisms, in Dynamical Systems, Peixoto (Ed.), Academic Press, New York, 1973, 493.Google Scholar
  6. [2.6]
    Shub, M. and Smale, S., Beyond hyperbolicity, Ann. of Math. 96 (1972), 587.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [2.7]
    Smale, S., Morse inequalities for a dynamical system, Bull. Amer. Math. Soc. 66 (1960), 43.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

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

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