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April  2005, 12(3): 403-412. doi: 10.3934/dcds.2005.12.403

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, Canada
2. 

Department of Mathematics, Wichita State University, Wichita, Kansas, 67260, United States

Received  September 2003 Revised  August 2004 Published  December 2004

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.
Keywords: box dimension, stable sets., SRB measure, Hausdorff dimension, Hyperbolic sets.
Mathematics Subject Classification: 37C45, 37C40, 37DX.
Citation: Rasul Shafikov, Christian Wolf. Stable sets, hyperbolicity and dimension. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 403-412. doi: 10.3934/dcds.2005.12.403
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