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>

A spectral approach to quenched linear and higher-order response for partially hyperbolic dynamics

“…The claim that ‘there exists such that is -mixing whenever ’ is exactly the content of [20, Proposition 6] (as well as being an easy corollary of [14, Proposition 3.11]).…”

“…In fact, [20, Proposition 6] and [14, Proposition 3.11] tell us that the claim follows from (b) of (QR0), (QR3) and (QR4) with . Furthermore, upon examining these proofs, it is clear that something slightly stronger is true: in the setting of Theorem 3.6, for every , there exists such that, for all , …”

“…On the other hand, the literature on quenched response theory for RDSs is relatively small and has only recently become an active research topic. With a few notable exceptions, most results for random systems have focused on the continuity of the equivariant random measure [3, 8, 22, 26, 37], although some more generally apply to the continuity of the Oseledets splitting and Lyapunov exponents associated to the Perron–Frobenius operator cocycle of the RDS [11, 14]. Quenched linear and higher-order response results are, to the best of our knowledge, limited to [44], where quenched linear and higher-order response is proved for general RDSs of uniformly expanding maps, and to [20], wherein quenched linear response is proved for RDSs of Anosov maps near a fixed Anosov map.…”

Application of exponential tight differential operators to the solution of the infectious disease equation

“…In the solution of infectious disease equations, this method can capture the dynamic changes of disease transmission more accurately and provide a more reliable decision basis for public health departments [7]. In addition, in the exponential compact difference operator, the design of the infectious disease equation solution structure is more flexible and changeable, itself has strong application reliability, to some extent, can provide a more accurate and efficient numerical solution method, to support the control and prevention of infectious diseases and prediction, promote the exponential compact differential operator method in other fields of application and development, improve the performance of the solution model, contribute to the development of the field of scientific computing [8].…”