Ergodic measures with large entropy have long unstable manifolds for $C^\infty$ surface diffeomorphisms

Abstract

We prove that for ergodic measures with large entropy have long unstable manifolds for $C^\infty$ surface diffeomorphisms. Specifically, for any $\alpha>0$, there exist constants $\beta>0$ and $c>0$ such that for every ergodic measure $\mu$ with metric entropy large than $\alpha$, the set of points with the size of unstable manifolds large than $\beta$ has $\mu$-measure large than $c$.