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 withnonzero Lyapunov exponents, then $F$ has ergodic measures of arbitrary intermediate entropies. To construct these measureswe find a set on which the return map is a skew product with horseshoesalong fibers. We can control the average return time and show the maximalentropy of these measures can be arbitrarily close to $h_\mu(F)$.
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