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 Gouëzel, 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 C k expanding maps on S 1 (k ≥ 2) and provide conditions for the stability of Lyapunov exponents and Oseledets splitting of the associated Perron-Frobenius operator cocycle to (i) uniformly small fiberwise C k−1 -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 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 Gouëzel-Keller-Liverani perturbation theory.

The unpredictability of chaotic nonlinear dynamics leads naturally to statistical descriptions, including probabilistic limit laws such as the central limit theorem and large deviation principle. A key tool in the Nagaev-Guivarc'h spectral method for establishing statistical limit theorems is a "twisted" transfer operator. In the abstract setting of Keller-Liverani [30] we prove that derivatives of all orders of the leading eigenvalues and eigenprojections of the twisted transfer operators with respect to the twist parameter are stable when subjected to a broad class of perturbations. As a result, we demonstrate stability of the variance in the central limit theorem and the rate function from a large deviation principle with respect to deterministic and stochastic perturbations of the dynamics and perturbations induced by numerical schemes. We apply these results to piecewise expanding maps in one and multiple dimensions, including new convergence results for Ulam projections on quasi-Hölder spaces.

We study the stability of statistical properties of Anosov maps on tori by examining the stability of the spectrum of an analytically twisted Perron-Frobenius operator on the anisotropic Banach spaces of Gouëzel and Liverani (2006 Ergod. Theor. Dyn. Syst. 26 189-217). By extending our previous work in Crimmins and Froyland (2019 Ann. Henri Poincaré 20 3113-3161), we obtain the stability of various statistical properties (the variance of a CLT and the rate function of an LDP) of Anosov maps to general perturbations, including new classes of numerical approximations. In particular, we obtain new results on the stability of the rate function under deterministic perturbations. As a key application, we focus on perturbations arising from numerical schemes and develop two new Fourier-analytic methods for efficiently computing approximations of the aforementioned statistical properties. This includes the first example of a rigorous scheme for approximating the peripheral spectral data of the Perron-Frobenius operator of an Anosov map without mollification. We consequently obtain the first rigorous numerical methods for estimating the variance and rate function for Anosov maps.

For smooth random dynamical systems we consider the quenched linear and higherorder response of equivariant physical measures to perturbations of the random dynamics. We show that the spectral perturbation theory of Gouëzel, Keller, and Liverani [36,32], which has been applied to deterministic systems with great success, may be adapted to study random systems that possess good mixing properties. As a consequence, we obtain general linear and higher-order response results for random dynamical systems that we then apply to random Anosov diffeomorphisms and random U(1) extensions of expanding maps. We emphasise that our results apply to random dynamical systems over a general ergodic base map, and are obtained without resorting to infinite dimensional multiplicative ergodic theory.

<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>