On the fractional susceptibility function of piecewise expanding maps

We associate to a perturbation (ft) of a (stably mixing) piecewise expanding unimodal map f0 a two-variable fractional susceptibility function Ψ_φ(η, z), depending also on a bounded observable φ. For fixed η ∈ (0, 1), we show that the function Ψ_φ(η, z) is holomorphic in a disc Dη ⊂ 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 7→ R_φ(t):= R φ(x) dµt, at t = 0, where µt is the unique absolutely continuous invariant probability measure of ft. In addition, we show that Ψ_φ(η, z) admits a holomorphic extension to the domain {(η, z) ∈ C² | 0 < <η < 1, z ∈ Dη }. Finally, if the perturbation (ft) is horizontal, we prove that lim_η∈_(0,1),η→₁ Ψ_φ(η, 1) = ∂tR_φ(t)|_t=0.