Measures of intermediate entropies for skew product diffeomorphisms

Peng Sun

doi:10.3934/dcds.2010.27.1219

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2010, Volume 27, Issue 3: 1219-1231. Doi: 10.3934/dcds.2010.27.1219

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Measures of intermediate entropies for skew product diffeomorphisms

Peng Sun^1,

1.
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802

Received: July 31, 2009

Revised: December 31, 2009

Published: August 2010

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Abstract

In this paper we study a skew product map $F$ preserving an ergodic measure $\mu$ of positive entropy. We show that if on the fibers the map are $C^{1+\alpha}$ diffeomorphisms with nonzero Lyapunov exponents, then $F$ has ergodic measures of arbitrary intermediate entropies. To construct these measures we find a set on which the return map is a skew product with horseshoes along fibers. We can control the average return time and show the maximal entropy of these measures can be arbitrarily close to $h_\mu(F)$.

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Mathematics Subject Classification: Primary: 37D25, 37C40; Secondary: 37A05.

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