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 ${\Psi }_{\varphi }(\eta ,z)$, depending also on a bounded observable $\varphi$. For fixed $\eta \in (0,1)$, we show that the function ${\Psi }_{\varphi }(\eta ,z)$ is holomorphic in a disc $D_{\eta }\subset \mathbb{C}$ centered at zero of radius $>1$, and that ${\Psi }_{\varphi }(\eta ,1)$ is the Marchaud fractional derivative of order $\eta$ of the function $t\mapsto {\mathcal{R}}_{\varphi }(t):=\int \varphi (x)d{\mu }_t$, at $t=0$, where ${\mu }_t$ is the unique absolutely continuous invariant probability measure of $f_t$. In addition, we show that ${\Psi }_{\varphi }(\eta ,z)$ 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 ${\lim }_{\eta \to 1}{\Psi }_{\varphi }(\eta ,1)={\partial }_t{\mathcal{R}}_{\varphi }(t){|}_{t=0}$.

arXiv commentVersion v2 is the electronic copy of the version published in DCDS

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• On convergence of linear response formulae in some piecewise hyperbolic maps
• Pre-threshold fractional susceptibility functions at Misiurewicz parameters