Diffeomorphisms with various $C^1$ stable properties

Details Diffeomorphisms with various $C^1$ stable properties Diffeomorphisms with various $C^1$ stable properties

2010-10-24

arxiv.org/abs/1010.4937v2

Acta Mathematica Scientia 2012,32B(2):552-558

Mathematics

Dynamical Systems

8 pages

Scientific paper

Let $M$ be a smooth compact manifold and $\Lambda$ be a compact invariant set. In this paper we prove that for every robustly transitive set $\Lambda$, $f|_\Lambda$ satisfies a $C^1-$generic-stable shadowable property (resp., $C^1-$generic-stable transitive specification property or $C^1-$generic-stable barycenter property) if and only if $\Lambda$ is a hyperbolic basic set. In particular, $f|_\Lambda$ satisfies a $C^1-$stable shadowable property (resp., $C^1-$stable transitive specification property or $C^1-$stable barycenter property) if and only if $\Lambda$ is a hyperbolic basic set. Similar results are valid for volume-preserving case.

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