Finiteness of measures of maximal entropy for smooth saddle surface endomorphisms

TL;DR

This paper proves that smooth saddle surface endomorphisms with sufficiently high entropy have finitely many ergodic measures of maximal entropy, using geometric constructions and Yomdin theory.

Contribution

It establishes finiteness of measures of maximal entropy for a class of smooth surface endomorphisms, extending understanding of their ergodic properties.

Findings

Finiteness of ergodic measures of maximal entropy under certain conditions

Construction of geometrically nice Markov rectangles

Application of Yomdin theory to analyze unstable curves

Abstract

We show that $\mathcal{C}^{\infty}$ local diffeomorphisms of closed surfaces whose topological entropy is larger than the logarithm of their degree admit a finite number of ergodic measures of maximal entropy. To do this, we construct families of rectangles, with a nice geometry, displaying a Markov property. We then analyze the behavior of the iterates of unstable curves intersecting these rectangles, using Yomdin theory.