Linear response for intermittent maps with critical point

TL;DR

This paper studies how the statistical properties of a family of intermittent maps with a critical point change smoothly with parameters, providing a formula for the derivative of invariant measure integrals with respect to these parameters.

Contribution

It extends linear response theory to a class of maps with neutral fixed points and critical points, using cone techniques to establish differentiability of invariant measure averages.

Findings

Proves differentiability of the map from parameters to invariant measure integrals.

Provides an explicit formula for the directional derivative in parameter space.

Extends linear response results to maps with critical points and neutral fixed points.

Abstract

We consider a two-parameter family of maps $T_{\alpha, \beta}: [0,1] \to [0,1]$ with a neutral fixed point and a non-flat critical point. Building on a cone technique due to Baladi and Todd, we show that for a class of $L^q$ observables $\phi: [0,1] \to \mathbb{R}$ the bivariate map $(\alpha, \beta) \mapsto \int_0^1 \phi \, d\mu_{\alpha,\beta}$, where $\mu_{\alpha, \beta}$ denotes the invariant SRB measure, is differentiable in a certain parameter region, and establish a formula for its directional derivative.