MUTIFRACTAL ANALYSIS FOR POINTWISE HOLDER EXPONENTS OF THE COMPLEX TAKAGI FUNCTIONS IN RANDOM COMPLEX DYNAMICS (The Theory of Random Dynamical Systems and Its Applications)

Abstract

We consider hyperbolic random complex dynamical systems on the Riemann sphere with separating condition and multiple rmnimal 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 todetermine the spectrum of pointwise Hölder exponents by employing the multifractal formalism in ergodic theory. Also, we prove that the bottom of the spectrum $alpha$_{-} is strictly less than 1, which allows us to show that the averaged system acts chaotically on the Banach space C^{ $alpha$} of $alpha$-Hölder continuous functions for every $alpha$in($alpha$_{-}, 1) , though the averaged system behaves very mildly (e.g. we have spectral gaps) on C^{ $beta$} for small $beta$>0.