Maximal measure and entropic continuity of Lyapunov exponents for $C^r$ surface diffeomorphisms with large entropy

TL;DR

This paper establishes the entropic continuity of Lyapunov exponents for smooth surface diffeomorphisms and shows the existence of measures of maximal entropy under certain entropy conditions.

Contribution

It provides a finite smooth version of entropic continuity for Lyapunov exponents and proves the existence of maximal entropy measures for $C^r$ surface diffeomorphisms with large entropy.

Findings

Finite smooth entropic continuity of Lyapunov exponents established.

Existence of maximal entropy measure under specified entropy conditions.

Continuity of topological entropy at the given diffeomorphism proved.

Abstract

We prove a finite smooth version of the entropic continuity of Lyapunov exponents of Buzzi-Crovisier-Sarig for $C^\infty$ surface diffeomorphisms [9]. As a consequence we show that any $C^r$, $r > 1$, smooth surface diffeomorphism $f$ with $h_{top}(f) > \frac{1}{r} \limsup_n \frac{1}{n} \log^+ \|df^n\|$ admits a measure of maximal entropy. We also prove the $C^r$ continuity of the topological entropy at $f$.