Pointwise Hölder exponents of the complex analogues of the Takagi function in random complex dynamics

We consider hyperbolic random complex dynamical systems on the Riemann sphere with separating condition and multiple minimal sets. We investigate the Hölder regularity of the function T of the probability of tending to one minimal set, the partial derivatives of T with respect to the probability parameters, which can be regarded as complex analogues of the Takagi function, and the higher partial derivatives C of T. Our main result gives a dynamical description of the pointwise Hölder exponents of T and C, which allows us to determine the spectrum of pointwise Hölder exponents by employing the multifractal formalism in ergodic theory. Also, we prove that the bottom of the spectrum α− is strictly less than 1, which allows us to show that the averaged system acts chaotically on the Banach space Cα of α-Hölder continuous functions for every α∈(α−,1), though the averaged system behaves very mildly (e.g. we have spectral gaps) on Cβ for small β>0.