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

TL;DR

This paper proves that for smooth surface diffeomorphisms, ergodic measures with high entropy are associated with long unstable manifolds, indicating a strong link between entropy and geometric instability.

Contribution

It establishes a quantitative relationship between high entropy measures and the length of unstable manifolds in $C^ abla$ surface diffeomorphisms, a novel connection in dynamical systems.

Findings

High entropy measures have large unstable manifolds

A positive measure set of points with large unstable manifolds exists

The measure of points with large unstable manifolds exceeds a positive constant

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 $c$.