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

## TL;DR

This paper develops abstract results on the regularity of fixed points with respect to parameters, accounting for regularity loss, and applies these to nonlinear maps and linear response in expanding dynamical systems.

## Contribution

It introduces a generalized approach to fixed point regularity considering regularity loss and higher order differentiability, with applications to dynamical systems.

## Key findings

- Fixed point regularity results considering regularity loss.
- Application to linear response in expanding dynamics.
- Extension to higher order differentiability via n-graded families.

## 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 Gou\"ezel-Liverani weak spectral perturbation theorem.