Linear response formula for piecewise expanding unimodal maps

TL;DR

This paper proves that for certain piecewise expanding unimodal maps, the average response function is differentiable at a parameter, aligning with a resummation of a divergent series predicted by Ruelle's conjecture.

Contribution

It establishes differentiability of the average response function under tangency conditions and connects it to a resummation of a divergent series, confirming a conjecture.

Findings

R(t) is differentiable at zero under tangency conditions

The derivative matches a resummation of a divergent series

Supports Ruelle's conjecture on linear response

Abstract

The average R(t) of a smooth function with respect to the SRB measure of a smooth one-parameter family f_t of piecewise expanding interval maps is not always Lipschitz. We prove that if f_t is tangent to the topological class of f_0, then R(t) is differentiable at zero, and the derivative coincides with the resummation previously proposed by the first named author of the (a priori divergent) series given by Ruelle's conjecture.