Expansive measures for flows

TL;DR

This paper generalizes the concept of expansive measures from homeomorphisms to flows, establishing key properties and applications, including a new proof that continuous expansive flows cannot exist on surfaces.

Contribution

It introduces the notion of expansive measures for flows, extending prior work, and explores their properties and implications, such as invariance and characterization for expansive flows.

Findings

No singularities in the support of expansive measures

Expansive flows are characterized by specific measure properties

No continuous expansive flows exist on surfaces

Abstract

We extend the concept of expansive measure \cite{am} defined for homeomorphism to flows. We obtain some properties for such measures including abscense of singularities in the support, aperiodicity, expansivity with respect to time-$T$ maps, invariance under flow-equivalence, negligibleness of orbits, characterization of expansive measures for expansive flows and naturallity under suspensions. As an application we obtain a new proof of the well known fact that there are no continuous expansive flows on surfaces (e.g. \cite{hs}).