Fractional Susceptibility Functions for the Quadratic Family: Misiurewicz–Thurston Parameters

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