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arXiv · 2510.04475

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

Matheus M. Castro·Gary Froyland
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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.

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Matheus M. Castro, Gary Froyland. 2025-10-06. On the cardinality of measures of maximal relative entropy for smooth skew products. https://arxiv.org/abs/2510.04475

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