Non-stationary compositions of Anosov diffeomorphisms

TL;DR

This paper investigates non-stationary compositions of Anosov diffeomorphisms, showing that such systems rapidly lose memory of initial states despite the absence of invariant measures, with implications for understanding non-equilibrium phenomena.

Contribution

It introduces a framework for analyzing non-stationary chaotic systems and proves exponential convergence of distributions without invariant measures.

Findings

Distributions converge exponentially over time

Systems lose statistical memory rapidly

No invariant measure exists for these systems

Abstract

Motivated by non-equilibrium phenomena in nature, we study dynamical systems whose time-evolution is determined by non-stationary compositions of chaotic maps. The constituent maps are topologically transitive Anosov diffeomorphisms on a 2-dimensional compact Riemannian manifold, which are allowed to change with time - slowly, but in a rather arbitrary fashion. In particular, such systems admit no invariant measure. By constructing a coupling, we prove that any two sufficiently regular distributions of the initial state converge exponentially with time. Thus, a system of the kind loses memory of its statistical history rapidly.