Attracting measures

TL;DR

This paper explores the properties of SRB measures on Axiom A attractors, emphasizing the importance of measure convergence from Lebesgue-absolutely continuous measures, and provides a topological abstraction of these properties.

Contribution

It introduces a topological abstraction of the convergence properties of SRB measures, highlighting the significance of measure pushforward convergence.

Findings

Empirical measures near attractors converge to SRB measures.

Pushforward of Lebesgue-absolutely continuous measures converges to SRB measures.

Provides a topological framework for understanding measure convergence.

Abstract

Under mild assumptions, the SRB measure $\mu$ associated to an Axiom A attractor $A$ has the following properties: (i) the empirical measure starting at a typical point near $A$ converges weakly to $\mu$; (ii) the pushforward of any Lebesgue-absolutely continuous probability measure supported near $A$ converges weakly to $\mu$. In general, a measure with the first property is called a "physical measure", and physical measures are recognised as generally important in their own right. In this paper, we highlight the second property as also important in its own right, and we prove a result that serves as a topological abstraction of the original result that establishes the second property for SRB measures on Axiom A attractors.