On the Invariance of Expansive Measures for Flows

TL;DR

This paper characterizes expansive measures for continuous flows on compact spaces, linking positive entropy to positive expansiveness and analyzing measure-theoretic properties of stable classes and measure sets.

Contribution

It introduces a new Borel set-based characterization of expansive measures and extends results relating positive entropy to positive expansiveness in flows.

Findings

Every ergodic invariant measure with positive entropy is positively expansive.

Flows with positive topological entropy have expansive invariant measures.

The set of expansive measures is a $G_{\delta\sigma}$-subset of all probability measures.

Abstract

We study expansive measures for continuous flows without fixed points on compact metric spaces. We provide a new characterization of expansive measures through dynamical balls that, in contrast to the dynamical balls considered in [\emph{J. Differ. Equ.}, 256 (2014):2246--2260], are actually Borel sets. This makes the theory more amenable to measure-theoretic analysis. We prove that every ergodic invariant measure with positive entropy is positively expansive, extending the results of \emph{Ergod. Th. \& Dynam. Sys.} \textbf{4}(3) (2014):765--776] to the setting of flows. This implies that flows with positive topological entropy admit expansive invariant measures. Furthermore, we show that the stable classes of such measures have zero measure. Lastly, we prove that the set of expansive measures for a flow is a $G_{\delta\sigma}$-subset of the space of all probability measures and that every expansive measure (invariant or not) can be approximated by expansive measures supported on invariant sets.