On the fractional susceptibility function of piecewise expanding maps

<p style='text-indent:20px;'>We associate to a perturbation <inline-formula><tex-math id="M1">\begin{document}$ (f_t) $\end{document}</tex-math></inline-formula> of (stably mixing) piecewise expanding unimodal map id="M2">\begin{document}$ f_0 two-variable fractional susceptibility function id="M3">\begin{document}$ \Psi_\phi(\eta, z) $\end{document}</tex-math></inline-formula>, depending also on bounded observable id="M4">\begin{document}$ \phi $\end{document}</tex-math></inline-formula>. For fixed id="M5">\begin{document}$ \eta \in (0,1) we show that the id="M6">\begin{document}$ is holomorphic in disc id="M7">\begin{document}$ D_\eta\subset \mathbb{C} centered at zero radius id="M8">\begin{document}$ >1 and id="M9">\begin{document}$ 1) Marchaud derivative order id="M10">\begin{document}$ id="M11">\begin{document}$ t\mapsto \mathcal{R}_\phi(t): = \int \phi(x)\, d\mu_t id="M12">\begin{document}$ t 0 where id="M13">\begin{document}$ \mu_t unique absolutely continuous invariant probability measure id="M14">\begin{document}$ f_t In addition, id="M15">\begin{document}$ admits extension domain id="M16">\begin{document}$ \{\, (\eta, \mathbb{C}^2\mid 0<\Re <1, \, z D_\eta \,\} Finally, if id="M17">\begin{document}$ horizontal, prove id="M18">\begin{document}$ \lim_{\eta (0,1), \to 1}\Psi_\phi(\eta, \partial_t \mathcal{R}_\phi(t)|_{t 0} $\end{document}</tex-math></inline-formula>.</p>