Pesin theory serves as a cornerstone of differentiable ergodic theory. A central result in Pesin theory states that for any ergodic hyperbolic measure, almost every point possesses an unstable manifold. However, the size of these unstable manifolds varies depending on the underlying measure.

Surface diffeomorphisms hold a particularly important position in the study of differentiable dynamical systems, as surfaces are the lowest-dimensional setting in which sufficient complexity emerges to illustrate the intricacies of diffeomorphisms. In this work, we establish that for $C^\infty$ surface diffeomorphisms, if the metric entropy of an ergodic measure is large, then there exists a set of positive measure such that every point in this set has an unstable manifold of uniformly large length. This uniformity depends on the entropy but is independent of the specific measure. This is joint work with Chiyi Luo.