Differentiability of the diffusion coefficient for a family of intermittent maps

TL;DR

This paper demonstrates that for a family of intermittent maps, the variance of the limiting normal distribution varies smoothly with the parameter when considering $C^2$ observables, extending previous results on correlation decay.

Contribution

It establishes the differentiability of the variance with respect to the parameter for a class of intermittent maps using Green-Kubo and linear response techniques.

Findings

Variance is a $C^1$ function of the parameter.

Differentiability shown for the first return map and extended to the original map.

Results apply to $C^2$ observables in the intermittent map family.

Abstract

It is well known that the Liverani-Saussol-Vaienti map satisfies a central limit theorem for H\"older observables in the parameter regime where the correlations are summable. We show that when $C^2$ observables are considered, the variance of the limiting normal distribution is a $C^1$ function of the parameter. We first show this for the first return map to the base of the second branch by studying the Green-Kubo formula, then conclude the result for the original map using Kac's lemma and relying on linear response.