On the Expansiveness of Invariant Measures under Pseudogroups

TL;DR

This paper introduces and analyzes weak and strong expansive measures for pseudogroups, exploring their properties, relations, and conditions under which they exist, including entropy criteria and specific cases like equicontinuous pseudogroups.

Contribution

It defines new notions of expansive measures for pseudogroups, establishes their relationships, and provides entropy-based criteria for their characterization, extending previous work.

Findings

Weak expansive measures can be characterized by positive entropy.

Equicontinuous pseudogroups may lack expansive measures.

Entropy positivity implies weak expansiveness.

Abstract

In this paper, we define and study weak expansive and expansive measures for pseudogroups, these two notions appear when analyzing the role of the generating set. We investigate the relations between such properties. We also provide a criterion for a measure to be weak expansive through the positivity of its entropy, generalizing the work of Arbieto and Morales. We also show that in some settings equicontinuous pseudogroups have no expansive measures.