On supports of expansive measures

TL;DR

This paper characterizes when a homeomorphism on a compact metric space admits expansive measures and explores the density and support properties of such measures, revealing structural implications for the space.

Contribution

It establishes an equivalence between the existence of expansive measures and many invariant supports, and analyzes the density of expansive measures in the space of probability measures.

Findings

Homeomorphisms with expansive measures have many invariant supports.

Expansive measures are dense in the space of Borel probability measures for certain homeomorphisms.

Spaces supporting these measures lack isolated points and have no interior in their heteroclinic sets.

Abstract

We prove that a homeomorphism of a compact metric space has an expansive measure \cite{ms} if and only if it has many ones with invariant support. We also study homeomorphisms for which the expansive measures are dense in the space of Borel probability measures. It is proved that these homeomorphisms exhibit a dense set of Borel probability measures which are expansive with full support. Therefore, their sets of heteroclinic points has no interior and the spaces supporting them have no isolated points.