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A Note on Generic Properties of Continuous Maps

Chapter
Part of the Progress in Mathematics book series (PM, volume 21)

Abstract

Let M be a compact manifold, with or without boundary. The genericity theorem of J. Palis, C. Pugh, M. Shub and D. Sullivan [PPSS] asserts that, among others, the property \( \Omega = \overline {\text{P}} \) (the set of non-wandering points is the closure of the set of periodic points) is C0-generic, i.e., holds for all homeomorphisms in some residual subset of the space Homeo(M) of all homeomorphisms of M to itself. This note points out and corrects a technical error in their proof, and extends the result to the space C0 (M, M) of all continuous maps of M to itself.

Keywords

Periodic Point Open Ball Compact Manifold Genericity Theorem Technical Error 
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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References

  1. [GP]
    V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, Englewood Cliffs, N.J., 1974.zbMATHGoogle Scholar
  2. [NS]
    Z. Nitecki and M. Snub, Filtrations, decompositions and explosions, Amer. J. Math. 97 (1976), 1029–1047.zbMATHCrossRefGoogle Scholar
  3. [PPSS]
    J. Palis, C. Pugh, M. Shub and D. Sullivan, Genericity theorems in topological dynamics, Dynamical Systems-Warwick 1974, pp. 241–250. Lectures Notes in Math., vol. 468, Springer, Berlin, 1975.Google Scholar
  4. [P]
    G. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math. 89 (1967), 1010–1021.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [S1]
    M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91. (1969), 175–199.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [S2]
    M. Shub, Stabilité globale des systèmes dynamiques, Astérisque, No. 56, Soc. Math. France, Paris, 1978.Google Scholar
  7. [SS]
    M. Shub and S. Smale, Beyond hyperbolicity, Ann. of Math. (2) 96 (1972), 587–591.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1982

Authors and Affiliations

  1. 1.Wesleyan UniversityMiddletownUSA
  2. 2.Tufts UniversityMedfordUSA

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