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 $h_{top}(f)>\frac{1}{r}{\mathrm{limsup}}_n\frac{1}{n}{\log }^{+}\Vert d{f}^n\Vert$ admits a measure of maximal entropy. We also prove the $C^r$ continuity of the topological entropy at $f$.

arXiv commentto appear in Annales Henri Poincar{é}