Upper semi-continuity of metric entropy for ${\mathcal{C}}^{1,\alpha }$ diffeomorphisms

Abstract

We establish a uniform approximation of metric entropy by partition entropy using uniform partitions for $\cC^{1,α}$ three-dimensional diffeomorphisms. This gives several consequences for diffeomorphisms on a compact manifold $M$ with $\dim M\le 3$. First, if an invariant measure $\mu$ is a continuity point of the sum of its positive Lyapunov exponents, then $\mu$ is an upper semi-continuity point of the entropy map. Second, it provides a slight improvement of the necessary condition for strong positive recurrence of surface and three-dimensional diffeomorphisms. Third, it yields continuity of dimensions for measures of maximal entropy.

This version reorganizes the structure of the theorem statements and provides a detailed comparison with previously known results