Stability of hyperbolic Oseledets splittings for quasi-compact operator cocycles

Harry Crimmins

doi:10.3934/dcds.2022001

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2022, Volume 42, Issue 6: 2795-2857. Doi: 10.3934/dcds.2022001

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Stability of hyperbolic Oseledets splittings for quasi-compact operator cocycles

Harry Crimmins

School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia

Received: May 31, 2020

Revised: August 31, 2021

Published: June 2022

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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 $ \mathcal{C}^k $ expanding maps on $ S^1 $ ($ k \ge 2 $) we provide conditions for the stability of Lyapunov exponents and Oseledets splitting of the cocycle under (ⅰ) uniformly small fiber-wise $ \mathcal{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 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.

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Mathematics Subject Classification: Primary: 37H15; Secondary: 37M25.

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