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

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  20 June 2023

HARRY CRIMMINS  and
YUSHI NAKANO
HARRY CRIMMINS*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia
YUSHI NAKANO
Affiliation:
Department of Mathematics, Tokai University, 4-1-1 Kitakaname, Hiratuka, Kanagawa, 259-1292, Japan (e-mail: yushi.nakano@tsc.u-tokai.ac.jp)
*
e-mail: hjc.gibbs@gmail.com
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Abstract

For smooth random dynamical systems we consider the quenched linear and higher-order response of equivariant physical measures to perturbations of the random dynamics. We show that the spectral perturbation theory of Gouëzel, Keller and Liverani [28, 33], 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, as well as the differentiability of the variance in quenched central limit theorems (CLTs), for random dynamical systems (RDSs) that we then apply to random Anosov diffeomorphisms and random U(1) extensions of expanding maps. We emphasize 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.

Keywords

linear responserandom dynamical systemspartially hyperbolic dynamics

MSC classification

Secondary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 4 , April 2024 , pp. 1026 - 1057
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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