Entropy and semiconjugacy on surfaces

Abstract

Let $g$ be a $C^{\infty }$ diffeomorphism in the isotopy class of a pseudo-Anosov homeomorphism $f$ such that $g$ and $f$ have the same topological entropy. In 1988, Handel proved that this implies the existence of a semiconjugacy $\pi$ from $g$ to $f$. He stated that, in general, there is at least one point $x$ such that ${\pi }^{-1}(x)$ is disconnected. We show that this is not the case: for every $x$, the set ${\pi }^{-1}(x)$ is the intersection of a nested sequence of closed topological disks, and hence is connected. We also prove that there is a unique $g$-invariant probability measure projecting to the measure of maximal entropy of $f$. This measure is entropy-maximizing, hyperbolic, and Bernoulli, and the semiconjugacy induces a metric isomorphism between the corresponding measure-preserving systems.