Entropies and volume growth of unstable manifolds

TL;DR

This paper explores the relationship between entropies along unstable manifolds, Lyapunov exponents, and volume growth in dynamical systems, providing new bounds on metric entropy and discussing extensions to less smooth cases.

Contribution

It establishes a novel connection between unstable foliation entropies and covering numbers, leading to improved bounds on metric entropy for $C^2$ and $C^{1+eta}$ diffeomorphisms.

Findings

Derived a new upper bound on metric entropy using Lyapunov exponents and volume growth.

Linked entropies along unstable manifolds to covering numbers and topological entropy.

Extended results to $C^{1+eta}$ diffeomorphisms.

Abstract

Let $f$ be a $C^2$ diffeomorphism on a compact manifold. Ledrappier and Young introduced entropies along unstable foliations for an ergodic measure $\mu$. We relate those entropies to covering numbers in order to give a new upper bound on the metric entropy of $\mu$ in terms of Lyapunov exponents and topological entropy or volume growth of sub-manifolds. We also discuss extensions to the $C^{1+\alpha},\,\alpha>0$ case.