Expansivity of ergodic measures with positive entropy

TL;DR

This paper demonstrates that ergodic measures with positive entropy assign zero measure to small dynamical neighborhoods, and explores implications for stable classes, topological entropy, and Li-Yorke chaos.

Contribution

It establishes a measure-zero property for dynamical neighborhoods under positive entropy measures and links this to stability and chaos properties.

Findings

Dynamical δ-balls have measure zero for ergodic measures with positive entropy.

Stable classes have measure zero under such measures.

Maps with countably many stable classes or Lyapunov stability have zero topological entropy.

Abstract

We prove that for every ergodic invariant measure with positive entropy of a continuous map on a compact metric space there is $\delta>0$ such that the dynamical $\delta$-balls have measure zero. We use this property to prove, for instance, that the stable classes have measure zero with respect to any ergodic invariant measure with positive entropy. Moreover, continuous maps which either have countably many stable classes or are Lyapunov stable on their recurrent sets have zero topological entropy. We also apply our results to the Li-Yorke chaos.