Stability of hyperbolic Oseledets splittings for quasi-compact operator cocycles

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