Metastability of Certain Intermittent Maps

TL;DR

This paper investigates how small perturbations in an intermittent map with two invariant densities cause the system to merge into one with a single invariant density, with convergence characterized by the sizes of the holes.

Contribution

It provides a rigorous analysis of the metastability and invariant density convergence in intermittent maps under perturbations involving holes.

Findings

Invariant density converges in $L^1$-norm to a convex combination of original densities.

The ratio of weights in the combination equals the limit of the ratio of hole measures.

Perturbations cause the system to transition from two to one invariant density.

Abstract

We study an intermittent map which has exactly two ergodic invariant densities. The densities are supported on two subintervals with a common boundary point. Due to certain perturbations, leakage of mass through subsets, called holes, of the initially invariant subintervals occurs and forces the subsystems to merge into one system that has exactly one invariant density. We prove that the invariant density of the perturbed system converges in the $L^1$-norm to a particular convex combination of the invariant densities of the intermittent map. In particular, we show that the ratio of the weights in the combination equals to the limit of the ratio of the measures of the holes.