On the fractional susceptibility function of piecewise expanding maps

Abstract

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

Pre-threshold fractional susceptibility functions at Misiurewicz parameters

“…) ∈ {±} and w 0 , w 1 > 0. Our goal here is to study the regularity of the η-Marchaud derivative, 0 ≤ η < 1 2 of ρ t 0 , for t 0 a Misiurewicz parameter. The main result of this section is: Theorem 9.…”

“…C] establishes the existence of some decomposition, related to the presence of poles on the unit circle, for the response and frozen fractional susceptibilities at the threshold value η = 1 2 . As a step towards proving those conjectures, we establish in Theorem 12, for mixing Misiurewicz parameters, holomorphy of ( 2), ( 3) and ( 5) in a disk of radius greater than one, for 0 ≤ η < 1 2 . In light of the previous discussion, this result is the best one can expect in this setting.…”

“…Rychlik and Sorets [7] showed that for yet another positive measure subset S ′′ of parameters, the invariant density ρ t ∈ L p for 1 ≤ p < 2 and that t ∈ S ′′ → ρ t ∈ L p , 1 ≤ p < 2 is continuous at Misiurewicz parameters, via a Hölder estimate. More recently, and for more general smooth unimodal families, Baladi, Benedicks and Schnellmann [3] proved that at almost every CE parameter t 0 , for any 1 2 -Hölder observable φ, the map R φ : t → φdµ t is C η at t = t 0 for any η < 1 2 , in the sense of Whitney, on a set of CE parameters having t 0 as a density point. Furthermore, they show that at any mixing MT parameter, there exists φ ∈ C ∞ , C > 1, and…”

“…In a recent preprint, Aspenberg, Baladi, Leppanen and Persson [1] introduced several fractional susceptibility functions, whose connection to fractional response is similar to the one between classical susceptibility function and linear response (i.e, the value at 1 of the (fractional) susceptibility function, if it is well-defined, is the (fractional) derivative of R φ at t = t 0 ). We recall here their definitions: given φ ∈ L ∞ (I t 0 ), and denoting L t the (Ruelle-Perron-Frobenius) transfer operator associated to f t , we define, as formal power series…”

“…To define a fractional susceptibility function in the spirit of [1], more care is needed. Indeed, as the invariant density ρ t is not defined for every parameter t, one has to consider integrals over some positive measure subset Ω ⊂ S. Then one sets…”