On the cardinality of measures of maximal relative entropy for smooth skew products

Abstract

Let $\Omega$ and $M$ be compact smooth manifolds and let $\Theta:\Omega\times M\to\Omega\times M$ be a $\mathcal C^{1+\alpha}$ skew-product diffeomorphism over a transitive Anosov base. We show that $\Theta$ has at most countably many ergodic hyperbolic measures of maximal relative entropy. When $\dim M=2$, if $\Theta$ has positive relative topological entropy, then $\Theta$ has at most countably many ergodic measures of maximal relative entropy.