Diffeomorphisms satisfying the specification property

Sakai, Kazuhiro; Sumi, Naoya; Yamamoto, Kenichiro

doi:10.1090/S0002-9939-09-10085-0

Diffeomorphisms satisfying the specification property
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by Kazuhiro Sakai, Naoya Sumi and Kenichiro Yamamoto PDF

Proc. Amer. Math. Soc. 138 (2010), 315-321 Request permission

Abstract:

Let $f$ be a diffeomorphism of a closed $C^\infty$ manifold $M$. In this paper, we introduce the notion of the $C^1$-stable specification property for a closed $f$-invariant set $\Lambda$ of $M$, and we prove that $f_{|\Lambda }$ satisfies a $C^1$-stable specification property if and only if $\Lambda$ is a hyperbolic elementary set. As a corollary, the $C^1$-interior of the set of diffeomorphisms of $M$ satisfying the specification property is characterized as the set of transitive Anosov diffeomorphisms.

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