Fractional susceptibility functions for the quadratic family: Misiurewicz-Thurston parameters

TL;DR

This paper introduces fractional susceptibility functions for the quadratic family, analyzing their singularities at specific parameters and proposing conjectures supported by numerical experiments to understand linear response phenomena.

Contribution

It defines fractional susceptibility functions for quadratic maps, studies their singularities at Misiurewicz-Thurston parameters, and formulates conjectures supported by numerical evidence.

Findings

Frozen susceptibility function has a pole at z=1 under certain conditions.

Conjecture that fractional susceptibility is singular at z=1 for Collet-Eckmann maps.

Numerical experiments support the formulated conjectures.

Abstract

For the quadratic family, we define the two-variable ($\eta$ and $z$) fractional susceptibility function associated to a C^1 observable at a stochastic map. We also define an approximate, "frozen" fractional susceptibility function. If the parameter is Misiurewicz-Thurston, we show that the frozen susceptibility function has a pole at $z=1$ for generic observables if a "one-half" transversality condition holds. We introduce "Whitney" fractional integrals and derivatives on suitable sets $\Omega$. We formulate conjectures supported by our results on the frozen susceptibility function and numerical experiments. In particular, we expect that the fractional susceptibility function for $\eta=1/2$ is singular at $z=1$ for Collet-Eckmann maps and generic observables. We view this work as a step towards the resolution of the paradox that the classical susceptibility function is holomorphic at $z=1$ for Misiurewicz-Thurston maps, despite lack of linear response.