The Diffusion Coefficient For Piecewise Expanding Maps Of The Interval With Metastable States

TL;DR

This paper studies how small perturbations in piecewise expanding interval maps with multiple invariant subintervals affect the diffusion coefficient of observables, approximating it via related Markov chains.

Contribution

It introduces a method to approximate the diffusion coefficient of perturbed maps using continuous-time Markov chains, linking invariant subintervals to a single invariant measure.

Findings

Approximate diffusion coefficients for perturbed maps.

Establishes a connection between deterministic maps and Markov chains.

Provides a practical method for analyzing metastable states.

Abstract

Consider a piecewise smooth expanding map of the interval possessing several invariant subintervals and the same number of ergodic absolutely continuous invariant probability measures (ACIMs). After this system is perturbed to make the subintervals lose their invariance in such a way that there is a unique ACIM, we show how to approximate the diffusion coefficient for an observable of bounded variation by the diffusion coefficient of a related continuous time Markov chain.