• Title/Summary/Keyword: weak measure expansive

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POSITIVELY WEAK MEASURE EXPANSIVE DIFFERENTIABLE MAPS

  • Ahn, Jiweon;Lee, Manseob
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.3
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    • pp.569-581
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    • 2020
  • In this paper, we introduce the new general concept of usual expansiveness which is called "positively weak measure expansiveness" and study the basic properties of positively weak measure expansive C1-differentiable maps on a compact smooth manifold M. And we prove that the following theorems. (1) Let 𝓟𝓦𝓔 be the set of all positively weak measure expansive differentiable maps of M. Denote by int(𝓟𝓦𝓔) is a C1-interior of 𝓟𝓦𝓔. f ∈ int(𝓟𝓦𝓔) if and only if f is expanding. (2) For C1-generic f ∈ C1 (M), f is positively weak measure-expansive if and only if f is expanding.

STABILITY OF WEAK MEASURE EXPANSIVE DIFFEOMORPHISMS

  • Ahn, Jiweon;Kim, Soyean
    • Journal of the Korean Mathematical Society
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    • v.55 no.5
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    • pp.1131-1142
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    • 2018
  • A notion of measure expansivity for homeomorphisms was introduced by Morales recently as a generalization of expansivity, and he obtained many interesting dynamic results of measure expansive homeomorphisms in [8]. In this paper, we introduce a concept of weak measure expansivity for homeomorphisms which is really weaker than that of measure expansivity, and show that a diffeomorphism f on a compact smooth manifold is $C^1$-stably weak measure expansive if and only if it is ${\Omega}$-stable. Moreover we show that $C^1$-generically, if f is weak measure expansive, then f satisfies both Axiom A and the no cycle condition.