Stability of hyperbolic Oseledets splittings for quasi-compact operator cocycles

TL;DR

This paper develops a framework for analyzing the stability of Oseledets splittings and Lyapunov exponents in random dynamical systems using a perturbation theory adapted to Perron-Frobenius operator cocycles, with applications to expanding maps on the circle.

Contribution

It introduces a random perturbation theory for Perron-Frobenius cocycles, incorporating Saks spaces, and applies it to stability and approximation problems in random expanding maps.

Findings

Established stability conditions for Lyapunov exponents under small perturbations.

Provided a numerical approximation method using Fejér kernels.

Extended the functional analytic approach with Saks spaces for dynamical systems.

Abstract

We consider the problem of stability and approximability of Oseledets splittings and Lyapunov exponents for Perron-Frobenius operator cocycles associated to random dynamical systems. By developing a random version of the perturbation theory of Gou\"ezel, Keller, and Liverani, we obtain a general framework for solving such stability problems, which is particularly well adapted to applications to random dynamical systems. We apply our theory to random dynamical systems consisting of $\mathcal{C}^k$ expanding maps on $S^1$ ($k \ge 2$) and provide conditions for the stability of Lyapunov exponents and Oseledets splitting of the associated Perron-Frobenius operator cocycle to (i) uniformly small fiber-wise $\mathcal{C}^{k-1}$-perturbations to the random dynamics, and (ii) numerical approximation via a Fej\'er kernel method. A notable addition to our approach is the use of Saks spaces, which provide a unifying framework for many key concepts in the so-called `functional analytic' approach to studying dynamical systems, such as Lasota-Yorke inequalities and Gou\"ezel-Keller-Liverani perturbation theory.