Pseudo-Orbits, Stationary Measures and Metastability

TL;DR

This paper investigates how small random perturbations affect multidimensional expanding maps, linking stationary measures to pseudo-orbits, and identifies conditions leading to metastable behavior in the system.

Contribution

It characterizes absolutely continuous stationary measures via pseudo-orbits and identifies metastability arising from specific perturbations.

Findings

Each least element has a neighborhood supporting a unique ergodic acsm.

Pseudo-orbits connect ergodic components of unperturbed systems.

Certain perturbations lead to metastable dynamics.

Abstract

We study random perturbations of multidimensional piecewise expanding maps. We characterize absolutely continuous stationary measures (acsm) of randomly perturbed dynamical systems in terms of pseudo-orbits linking the ergodic components of absolutely invariant measures (acim) of the unperturbed system. We focus on those components, called least-elements, which attract pseudo-orbits. We show that each least element admits a neighbourhood which supports exactly one ergodic acsm of the random system. We use this result to identify random perturbations that exhibit a metastable behavior.