On the fractional susceptibility function of piecewise expanding maps

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