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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 \cite{M}. 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.
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참고문헌 (13)
참고문헌 신청
A. Arbieto / 2011 / Periodic orbits and expansiveness / Math. Z. 269(3-4):801~807
A. Artigue / 2015 / A note on measure-expansive diffeomorphisms / J. Math. Anal. Appl. 428(1):713~716
J. Franks / 1971 / Necessary conditions for stability of diffeomorphisms / Trans. Amer. Math. Soc. 158:301~308
S. Hayashi / 1992 / Diffomorphisms in $F^1(M)$ satisfy Axiom A / Ergod. Th. & Dynam. Sys. 12:233~253
M. Lee / 2017 / Weak measure expansiveness for partially hyperbolic diffeomorphisms / Chaos Solitons Fractals 103:256~260
A. Artigue / 2015 / A note on measure-expansive diffeomorphisms / J. Math. Anal. Appl. 428(1):713~716
J. Franks / 1971 / Necessary conditions for stability of diffeomorphisms / Trans. Amer. Math. Soc. 158:301~308
S. Hayashi / 1992 / Diffomorphisms in
M. Lee / 2017 / Weak measure expansiveness for partially hyperbolic diffeomorphisms / Chaos Solitons Fractals 103:256~260
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