August  2010, 27(3): 1219-1231. doi: 10.3934/dcds.2010.27.1219

Measures of intermediate entropies for skew product diffeomorphisms

1. 

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States

Received  August 2009 Revised  January 2010 Published  March 2010

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 $h_\mu(F)$.
Citation: Peng Sun. Measures of intermediate entropies for skew product diffeomorphisms. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1219-1231. doi: 10.3934/dcds.2010.27.1219
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