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

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 $f$.