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

For f t ( x ) = t - x 2 the quadratic family, we define the fractional susceptibility function Ψ ϕ , t 0 Ω ( η , z ) of f t , associated to a C 1 observable ϕ at a stochastic parameter t 0 . We also define an approximate, “frozen,” fractional susceptibility function Ψ ϕ , t 0 fr ( η , z ) such that lim η → 1 Ψ ϕ , t 0 fr ( η , z ) is the susceptibility function Ψ ϕ , t 0 ( z ) studied by Ruelle. If t 0 is Misiurewicz–Thurston, we show that Ψ ϕ , t 0 fr ( 1 / 2 , z ) has a pole at z = 1 for generic ϕ if J 1 / 2 ( t 0 ) ≠ 0 , where J η ( t ) = ∑ k = 0 ∞ sgn ( D f t k ( c 1 ) ) | D f t k ( c 1 ) | - η , with c 1 = t the critical value of f t . We introduce “Whitney” fractional integrals I η , Ω and derivatives M η , Ω on suitable sets Ω . We formulate conjectures on Ψ ϕ , t 0 Ω ( η , z ) and J η ( t ) , supported by our results on M η , Ω and Ψ ϕ , t 0 fr ( 1 / 2 , z ) , for the former, and numerical experiments, for the latter. In particular, we expect that Ψ ϕ , t 0 Ω ( 1 / 2 , z ) is singular at z = 1 for Collet–Eckmann t 0 and generic ϕ . We view this work as a step towards the resolution of the paradox that Ψ ϕ , t 0 ( z ) is holomorphic at z = 1 for Misiurewicz–Thurston f t 0 (Jiang and Ruelle in Nonlinearity 18:2447–2453, 2005, Ruelle in Commun Math Phys 258:445–453, 2005), despite lack of linear response (Baladi et al. in Invent Math 201:773–844, 2015).