Stability of hyperbolic Oseledets splittings for quasi-compact operator cocycles

Abstract

<p style='text-indent:20px;'>We develop a random version of the perturbation theory of Gouëzel, Keller, and Liverani, and consequently obtain results on the stability of Oseledets splittings and Lyapunov exponents for operator cocycles. By applying the theory to the Perron-Frobenius operator cocycles associated to random <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{C}^k $\end{document}</tex-math></inline-formula> expanding maps on <inline-formula><tex-math id="M2">\begin{document}$ S^1 $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M3">\begin{document}$ k \ge 2 $\end{document}</tex-math></inline-formula>) we provide conditions for the stability of Lyapunov exponents and Oseledets splitting of the cocycle under (ⅰ) uniformly small fiber-wise <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{C}^{k-1} $\end{document}</tex-math></inline-formula>-perturbations to the random dynamics, and (ⅱ) numerical approximation via a Fejér kernel method. A notable addition to our approach is the use of Saks spaces, which allow us to weaken the hypotheses of Gouëzel-Keller-Liverani perturbation theory, provides a unifying framework for key concepts in the so-called 'functional analytic' approach to studying dynamical systems, and has applications to the construction of anisotropic norms adapted to dynamical systems.</p>