Generic Continuity of Metric Entropy for Volume-preserving Diffeomorphisms

Abstract

Let $M$ be a compact manifold and ${\text{Diff}}_m^1(M)$ be the set of $C^1$ volume-preserving diffeomorphisms of $M$. We prove that there is a residual subset $\mathcal{R}\subset {\text{Diff}}_m^1(M)$ such that each $f\in \mathcal{R}$ is a continuity point of the map $g\to h_m(g)$ from ${\text{Diff}}_m^1(M)$ to $\mathbb{R}$, where $h_m(g)$ is the metric entropy of $g$ with respect to volume measure $m$.