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

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