A regularity result for fixed points, with applications to linear response

Abstract

In this paper, we show a series of abstract results on fixed point regularity with respect to a parameter. They are based on a Taylor development taking into account a loss of regularity phenomenon, typically occurring for composition operators acting on spaces of functions with finite regularity. We generalize this approach to higher order differentiability, through the notion of an n-graded family. We then give applications to the fixed point of a non linear map, and to linear response in the context of (uniformly) expanding dynamics (theorem 3 and corollary 2), in the spirit of Gouëzel-Liverani weak spectral perturbation theorem.

25 pages; Accepted for publication in Nonlinearity; Revised version after referees reports

References in corpus (2)

• Linear response for intermittent maps
• Linear response, or else

Cited by in corpus (7)

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• Self consistent transfer operators. Invariant measures, convergence to equilibrium, linear response and control of the statistical properties
• Fréchet differentiable drift dependence of Perron--Frobenius and Koopman operators for non-deterministic dynamics
• Regularity of characteristic exponents and linear response for transfer operator cocycles
• Quenched linear response for smooth expanding on average cocycles
• A spectral approach to quenched linear and higher-order response for partially hyperbolic dynamics
• Regularity of the spectrum for expanding maps