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

TL;DR

This paper develops a framework for analyzing the differentiability of SRB measures in one-dimensional non-uniformly expanding maps, including systems with infinite measures, and derives a linear response formula applicable to both finite and infinite SRB measures.

Contribution

It introduces a novel inducing technique to transfer linear response properties from uniformly to non-uniformly expanding systems, covering cases with infinite SRB measures.

Findings

Proves differentiability of SRB measures for Pomeau-Manneville maps.

Derives a linear response formula for systems with infinite SRB measures.

First to establish linear response in infinite measure-preserving systems.

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.