A regularity result for fixed points, with applications to linear response. (5th March 2018)

• Record Type: Journal Article
• Authors: Sedro, Julien
• Is Part Of: Nonlinearity. Volume 31:Number 4(2018:Apr.)
• Page Start: 1417
• Page End: 1440
• Publication Date: 2018-03-05
• Subjects: dynamical systems -- fixed-point differentiability -- linear response -- expanding maps; 37C30 -- 47H10 -- 37F15; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aaa10b
• Languages: English
• ISSNs: 0951-7715

Abstract

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 nonlinear map, and to linear response in the context of (uniformly) expanding dynamics (theorem7 and corollary20 ), in the spirit of Gouëzel–Liverani.