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October 1994 An improvement of the $C^{1}$ closing lemma for endomorphisms
Hiroshi IKEDA
Hokkaido Math. J. 23(3): 427-437 (October 1994). DOI: 10.14492/hokmj/1381413097

Abstract

L. Wen proved the $C^{1}$ closing lemma for endomorphisms with finitely many singularities. The arguments and the tools of Wen are available for endomorphisms which have infinitely many singularities but at most finitely many ones in the nonwandering sets. By refining the argument of Wen we prove the $C^{1}$ closing lemma for endomorphisms with finitely many singularities in the nonwandering sets. By using this lemma we can slightly improve characterization of $C^{1}$ absolutely $\Omega$-stable endomorphisms. That is, for an endomorphism $f$ with finitely many singularities in the nonwandering set, $f$ is $C^{1}$ absolutely $\Omega$-stable if and only if $f$ has a neighborhood $\mathscr{U}$ such that every $g$ in $\mathscr{U}$ satisfies weak Axiom A.

Citation

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Hiroshi IKEDA. "An improvement of the $C^{1}$ closing lemma for endomorphisms." Hokkaido Math. J. 23 (3) 427 - 437, October 1994. https://doi.org/10.14492/hokmj/1381413097

Information

Published: October 1994
First available in Project Euclid: 10 October 2013

zbMATH: 0824.58013
MathSciNet: MR1299635
Digital Object Identifier: 10.14492/hokmj/1381413097

Subjects:
Primary: 58F20

Rights: Copyright © 1994 Hokkaido University, Department of Mathematics

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Vol.23 • No. 3 • October 1994
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