Maximal Measure and Entropic Continuity of Lyapunov Exponents for Cr Surface Diffeomorphisms with Large Entropy.

We prove a finite smooth version of the entropic continuity of Lyapunov exponents proved recently by Buzzi, Crovisier, and Sarig for C ∞ surface diffeomorphisms (Buzzi et al., Invent Math 230(2):767–849, 2022). As a consequence, we show that any C r , r > 1 , smooth surface diffeomorphism f with h top (f) > 1 r lim sup n 1 n log + ‖ d f n ‖ ∞ admits a measure of maximal entropy. We also prove the C r continuity of the topological entropy at f. [ABSTRACT FROM AUTHOR]