Stability of invariant measures

TL;DR

This paper introduces a new topology on the set of invariant measures in dynamical systems, generalizing stability notions and connecting to a limit case of a $p$-optimal transport problem, offering a novel perspective on measure stability.

Contribution

It defines a new topology on invariant measures that extends stability concepts and links to a $p$-optimal transport problem for $p=\infty$, providing a fresh mathematical framework.

Findings

The topology differs from weak* and Riesz topologies.

It is a solution to a limit case of a $p$-optimal transport problem.

The topology offers new insights into measure stability in dynamical systems.

Abstract

We generalize various notions of stability of invariant sets of dynamical systems to invariant measures, by defining a topology on the set of measures. The defined topology is similar, but not topologically equivalent to weak* topology, and it also differs from topologies induced by the Riesz Representation Theorem. It turns out that the constructed topology is a solution of a limit case of a $p$-optimal transport problem, for $p=\infty$.