Empiric stochastic stability of physical and pseudo-physical measures

TL;DR

This paper introduces the concept of empiric stochastic stability for invariant measures in dynamical systems, establishing its equivalence with physical measures and exploring properties of stable sets, especially in expanding systems.

Contribution

It defines empiric stochastic stability for sets of invariant measures and proves their relation to physical and pseudo-physical measures, extending stability analysis.

Findings

Empiric stochastic stability of a measure is equivalent to it being physical.

Stable sets of measures can exist without physical measures, composed of pseudo-physical measures.

In expanding systems, stable measures satisfy Pesin Entropy Formula.

Abstract

We define the empiric stochastic stability of an invariant measure in the finite-time scenario, the classical definition of stochastic stability. We prove that an invariant measure of a continuous system is empirically stochastically stable if and only if it is physical. We also define the empiric stochastic stability of a weak*-compact set of invariant measures instead of a single measure. Even when the system has not physical measures it still has minimal empirically stochastically stable sets of measures. We prove that such sets are necessarily composed by pseudo-physical measures. Finally, we apply the results to the one-dimensional C1-expanding case to conclude that the measures of empirically stochastically sets satisfy Pesin Entropy Formula.