Local stable and unstable sets for positive entropy $C^1$ dynamical systems

TL;DR

This paper establishes lower bounds on the Hausdorff dimension of local stable and unstable sets for positive entropy $C^1$ dynamical systems, linking geometric properties with entropy and Lyapunov exponents, and extends to infinite-dimensional systems.

Contribution

It provides a new dimension estimate for local stable and unstable sets in $C^1$ systems with positive entropy, applicable to both finite and infinite-dimensional cases.

Findings

Lower bounds on Hausdorff dimension in terms of entropy and Lyapunov exponents

Applicable to infinite-dimensional $C^1$ dynamical systems

Framework based on topological dynamical systems

Abstract

For any $C^1$ diffeomorphism on a smooth compact Riemannian manifold that admits an ergodic measure with positive entropy, a lower bound of the Hausdorff dimension for the local stable and unstable sets is given in terms of the measure-theoretic entropy and the maximal Lyapunov exponent. The mainline of our approach to this result is under the settings of topological dynamical systems, which is also applicable to infinite dimensional $C^1$ dynamical systems.