Regular maps with the specification property

Kazumine Moriyasu; Kazuhiro Sakai; Kenichiro Yamamoto

doi:10.3934/dcds.2013.33.2991

Article Contents

2013, Volume 33, Issue 7: 2991-3009. Doi: 10.3934/dcds.2013.33.2991

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Regular maps with the specification property

1.
Department of Mathematics, Tokushima University, Tokushima 770-8502
2.
Department of Mathematics, Utsunomiya University, Utsunomiya 321-8505
3.
School of Information Environment, Tokyo Denki University, 2-1200 Buseigakuendai, Inzai-shi, Chiba 270-1382

Received: February 29, 2012

Revised: July 31, 2012

Published: June 2013

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Abstract

Let $f$ be a $C^1$-regular map of a closed $C^{\infty}$ manifold $M$ and $\Lambda$ be a locally maximal closed invariant set of $f$. We show that $f|_{\Lambda}$ satisfies the $C^1$-stable specification property if and only if $\Lambda$ is a hyperbolic elementary set. We also prove that there exists a residual subset $\mathcal{R}$ in the space of $C^1$-regular maps endowed with the $C^1$-topology such that for $f \in \mathcal{R}$, $f|_{\Lambda}$ satisfies the specification property if and only if $\Lambda$ is a hyperbolic elementary set.

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Mathematics Subject Classification: Primary: 37A25, 37B35, 37C20, 37D20, 37D30.

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