Linear response for intermittent maps with summable and nonsummable decay of correlations

TL;DR

This paper studies a family of interval maps with varying decay rates of correlations, showing that certain statistical properties depend smoothly on parameters despite the lack of a spectral gap.

Contribution

It demonstrates the continuous differentiability of invariant measure integrals with respect to parameters for Pomeau-Manneville maps, even without a spectral gap and with nonsummable correlations.

Findings

Invariant measures depend smoothly on parameters.

Decay of correlations varies as a power law.

Differentiability holds despite absence of spectral gap.

Abstract

We consider a family of Pomeau-Manneville type interval maps $T_\alpha$, parametrized by $\alpha \in (0,1)$, with the unique absolutely continuous invariant probability measures $\nu_\alpha$, and rate of correlations decay $n^{1-1/\alpha}$. We show that despite the absence of a spectral gap for all $\alpha \in (0,1)$ and despite nonsummable correlations for $\alpha \geq 1/2$, the map $\alpha \mapsto \int \varphi \, d\nu_\alpha$ is continuously differentiable for $\varphi \in L^{q}[0,1]$ for $q$ sufficiently large.