Robustness of ergodic properties of nonautonomous piecewise expanding maps

TL;DR

This paper demonstrates the stochastic stability and robustness of ergodic properties in nonautonomous compositions of piecewise expanding maps, showing that their statistical behavior remains close to that of the unperturbed map.

Contribution

It extends stability results to more general perturbations of piecewise expanding maps and analyzes the behavior of distributions and Birkhoff averages under nonautonomous compositions.

Findings

Perturbed maps are stochastically stable.

Distributions under perturbations stay close to invariant measures.

Birkhoff averages do not fluctuate wildly for almost every point.

Abstract

Recently, there has been an increasing interest on nonautonomous composition of perturbed hyperbolic systems: composing perturbations of a given hyperbolic map $F$ results in statistical behaviour close to that of $F$. We show this fact in the case of piecewise regular expanding maps. In particular, we impose conditions on perturbations of this class of maps that include situations slightly more general than what has been considered so far, and prove that these are stochastically stable in the usual sense. We then prove that the evolution of a given distribution of mass under composition of time dependent perturbations (arbitrarily - rather than randomly - chosen at each step) close to a given map $F$ remains close to the invariant mass distribution of $F$. Moreover, for almost every point, Birkhoff averages along trajectories do not fluctuate wildly. This result complements recent results on memory loss for nonautonomous dynamical systems.