Approximating invariant densities of metastable systems

TL;DR

This paper studies how the invariant densities of a system with two separate regions can be approximated when the regions merge due to small perturbations, providing a method to understand the transition of invariant measures.

Contribution

It introduces a way to approximate the invariant density of a perturbed system with merged regions using a convex combination of the original densities.

Findings

The invariant density of the perturbed system can be approximated by a convex combination of original densities.

The approach applies to piecewise smooth expanding maps with two invariant subsets.

Provides insights into the behavior of invariant measures under small perturbations.

Abstract

We consider a piecewise smooth expanding map of the interval possessing two invariant subsets of positive Lebesgue measure and exactly two ergodic absolutely continuous invariant probability measures (ACIMs). When this system is perturbed slightly to make the invariant sets merge, we describe how the unique ACIM of the perturbed map can be approximated by a convex combination of the two initial ergodic ACIMs.