Stable sets, hyperbolicity and dimension

Rasul Shafikov; Christian Wolf

doi:10.3934/dcds.2005.12.403

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2005, Volume 12, Issue 3: 403-412. Doi: 10.3934/dcds.2005.12.403

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Stable sets, hyperbolicity and dimension

Rasul Shafikov^1, and
Christian Wolf^2,

1.
Department of Mathematics, Middlesex College, The University of Western Ontario, London, Ontario N6A 5B7
2.
Department of Mathematics, Wichita State University, Wichita, Kansas, 67260

Received: August 31, 2003

Revised: July 31, 2004

Published: March 2005

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Abstract

In this note we derive an upper bound for the Hausdorff and box dimension of the stable and local stable set of a hyperbolic set $\Lambda$ of a $C^2$ diffeomorphisms on a $n$-dimensional manifold. As a consequence we obtain that dim$_H W^s(\Lambda)=n$ is equivalent to the existence of a SRB-measure. We also discuss related results for expanding maps.

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Mathematics Subject Classification: 37C45, 37C40, 37DXX.

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