Expansive Measures versus Lyapunov exponents

TL;DR

This paper explores the relationship between measure expansiveness and hyperbolicity, establishing conditions under which positive measure-expansiveness implies expansion and hyperbolic structures in dynamical systems.

Contribution

It proves that invariant ergodic measures with all positive Lyapunov exponents are positively measure-expansive and links robust positive measure-expansiveness to expansion.

Findings

Positive measure-expansiveness for measures with all positive Lyapunov exponents

Robust positive measure-expansiveness implies expansion in local diffeomorphisms

Volume-preserving diffeomorphisms not approximable by positively measure-expansive ones have dominated splitting

Abstract

In this paper we investigate the relation between measure expansiveness and hyperbolicity. We prove that non atomic invariant ergodic measures with all of its Lyapunov exponents positive is positively measure-expansive. We also prove that local diffeomorphisms robustly positively measure-expansive is expanding. Finally, we prove that if a $C^1$ volume preserving diffeomorphism that. can not be accumulated by positively measure expansive diffeomorphis have a dominated sppliting.