Linear response in the intermittent family: Differentiation in a weighted $C^0$-norm

Benoît Saussol,Wael Bahsoun

doi:10.3934/dcds.2016089

Abstract

We provide a general framework to study differentiability of SRB measures for one dimensional non-uniformly expanding maps. Our technique is based on inducing the non-uniformly expanding system to a uniformly expanding one, and on showing how the linear response formula of the non-uniformly expanding system is inherited from the linear response formula of the induced one. We apply this general technique to interval maps with a neutral fixed point (Pomeau-Manneville maps) to prove differentiability of the corresponding SRB measure. Our work covers systems that admit a finite SRB measure and it also covers systems that admit an infinite SRB measure. In particular, we obtain a linear response formula for both finite and infinite SRB measures. To the best of our knowledge, this is the first work that contains a linear response result for infinite measure preserving systems.

Highlights

In physical applications of dynamical systems, it is important to understand how statistical properties of a perturbed physical system are related to statistical properties of the original system; i.e., before the occurrence of the perturbation
We introduce a class of interval maps which are non-uniformly expanding with two branches, for which one can construct an inducing scheme which allow to inherit the linear response formula from the one for the induced system
We introduce here a class of interval maps which are uniformly expanding, with a finite or countable number of branches, for which we will be able to prove a linear response formula

Summary

Introduction

In physical applications of dynamical systems, it is important to understand how statistical properties of a perturbed physical system are related to statistical properties of the original system; i.e., before the occurrence of the perturbation. In this work we provide a general framework to study differentiability of SRB measures for one dimensional non-uniformly expanding maps. We cover both the finite and infinite SRB measure cases and we prove differentiability in norm1.

Results

Conclusion