Linear response for intermittent maps

TL;DR

This paper establishes a linear response formula for the SRB measure of a family of intermittent maps with polynomial mixing rates, extending linear response theory to slowly mixing dynamical systems.

Contribution

It proves differentiability of the invariant measure's integral with respect to the parameter and derives a linear response formula in a setting with polynomial decay of correlations.

Findings

Linear response formula derived for intermittent maps.

Differentiability of invariant measure with respect to parameters proven.

Applicable to systems with polynomial mixing rates.

Abstract

We consider the one parameter family $\alpha \mapsto T_\alpha$ ($\alpha \in [0,1)$) of Pomeau-Manneville type interval maps $T_\alpha(x)=x(1+2^\alpha x^\alpha)$ for $x \in [0,1/2)$ and $T_\alpha(x)=2x-1$ for $x \in [1/2, 1]$, with the associated absolutely continuous invariant probability measure $\mu_\alpha$. For $\alpha \in (0,1)$, Sarig and Gou\"ezel proved that the system mixes only polynomially with rate $n^{1-1/\alpha}$ (in particular, there is no spectral gap). We show that for any $\psi\in L^q$, the map $\alpha \to \int_0^1 \psi\, d\mu_\alpha$ is differentiable on $[0,1-1/q)$, and we give a (linear response) formula for the value of the derivative. This is the first time that a linear response formula for the SRB measure is obtained in the setting of slowly mixing dynamics. Our argument shows how cone techniques can be used in this context. For $\alpha \ge 1/2$ we need the $n^{-1/\alpha}$ decorrelation obtained by Gou\"ezel under additional conditions.