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

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.

arXiv commentVersions v3-v4 contain the electronic copy of the published version in Comm Math Phys. 4 figures. Version v4 includes a 3-page long supplementary note

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• Pre-threshold fractional susceptibility functions at Misiurewicz parameters
• A parameter ASIP for the quadratic family