On the fractional susceptibility function of piecewise expanding maps

Abstract

We associate to a perturbation $(f_t)$ of a (stably mixing) piecewise expanding unimodal map $f_0$ a two-variable fractional susceptibility function $Ψ_ϕ(η, z)$, depending also on a bounded observable $ϕ$. For fixed $η\in (0,1)$, we show that the function $Ψ_ϕ(η, z)$ is holomorphic in a disc $D_η\subset \mathbb{C}$ centered at zero of radius $>1$, and that $Ψ_ϕ(η, 1)$ is the Marchaud fractional derivative of order $η$ of the function $t\mapsto \mathcal{R}_ϕ(t):=\int ϕ(x)\, dμ_t$, at $t=0$, where $μ_t$ is the unique absolutely continuous invariant probability measure of $f_t$. In addition, we show that $Ψ_ϕ(η, z)$ admits a holomorphic extension to the domain $\{ (η, z) \in {\mathbb{C}}^2\mid 0<\Re η<1, \, z \in D_η\}$. Finally, if the perturbation $(f_t)$ is horizontal, we prove that $\lim_{η\to 1}Ψ_ϕ(η, 1)=\partial_t \mathcal{R}_ϕ(t)|_{t=0}$.