Stability of hyperbolic Oseledets splittings for quasi-compact operator cocycles

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.