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On proofs of the $C^0$ general density theorem

Author: Mike Hurley
Journal: Proc. Amer. Math. Soc. 124 (1996), 1305-1309
MSC (1991): Primary 58F08
MathSciNet review: 1307531
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Abstract: We show that if $M$ is a compact manifold, then there is a residual subset ${\mathcal{N}}$ of the set of homeomorphisms on $M$ with the property that if $f\in {\mathcal{N}}$, then the periodic points of $f$ are dense in its chain recurrent set. This result was first announced in [4], but a flaw in that argument was noted in [1], where a different proof was given. It was recently noted in [5] that this new argument only serves to show that the density of periodic points in the chain recurrent set is generic in the closure of the set of diffeomorphisms. We show how to patch the original argument from [4] to prove the result.

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  • 1. E.M. Coven, J. Madden, Z. Nitecki, A note on generic properties of continuous maps, Ergodic Theory and Dynamical Systems II, Boston, Birkhäuser, 1982, pp. 97--101. MR 84c:58068
  • 2. J. Munkres, Obstructions to the smoothing of piecewise-differentiable homeomorphisms, Ann. of Math. 72 (1960), 521-554. MR 22:12534
  • 3. Z. Nitecki, M. Shub, Filtrations, decompositions, and explosions, Amer. J. Math. 97 (1976), 1029-1047.MR 52:15561
  • 4. J. Palis, C. Pugh, M. Shub, M. Sullivan, Genericity theorems in topological dynamics, Dynamical Systems -- Warwick 1974 (Springer Lect. Notes in Math. #468), Springer-Verlag, New York, 1975, pp. 241--250.MR 58:31268
  • 5. S.Y. Pilyugin, The Space of Dynamical Systems with the $C^0$ Topology, (Springer Lect. Notes in Math #1571), Springer-Verlag, New York, 1994.MR 37:2257
  • 6. C. C. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math. 89 (1967), 1010--1021.
  • 7. F. Takens, On Zeeman's tolerance stability conjecture, Manifolds -- Amsterdam 1970 (Springer Lect. Notes in Math #197), Springer-Verlag, New York, 1971, pp. 209--219.MR 43:5511

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Additional Information

Mike Hurley

Keywords: Chain recurrent set, generic homeomorphism
Received by editor(s): October 11, 1994
Communicated by: Mary Rees
Article copyright: © Copyright 1996 American Mathematical Society