Rescaled expansive measures for flows

TL;DR

This paper introduces rescaled expansive measures for flows, extending measure-theoretic concepts to include singularities, and establishes their properties and connections to entropy through new formulas.

Contribution

It develops a measure-theoretic framework for rescaled expansiveness in flows with singularities, including invariant measures and a generalized entropy formula.

Findings

Existence of invariant rescaled expansive measures under positive entropy.

Extension of the Brin-Katok local entropy formula to flows with singularities.

New tools for analyzing entropy and expansiveness in continuous-time systems.

Abstract

We introduce the notion of rescaled expansive measures to study a measure-theoretic formulation of rescaled expansiveness for flows, particularly in the presence of singularities. Equivalent definitions are established via reparametrizations of different regularities. Under the assumption of positive entropy, we prove the existence of invariant rescaled expansive measures. In the appendix, we derive a rescaled version of the Brin-Katok local entropy formula for flows, extending [7] from nonsingular flows to general flows that may include singularities. This framework provides new tools for understanding entropy and expansiveness in continuous-time dynamical systems with singularities.