Pointwise Hölder Exponents of the Complex Analogues of the Takagi Function in Random Complex Dynamics

Abstract

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 ${\alpha }_{-}$ 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 $\beta >0.$

Published in Adv. Math. 313 (2017) 839--874. See also http://www.math.shimane-u.ac.jp/~jaerisch/ and http://www.math.h.kyoto-u.ac.jp/~sumi/index.html

References in corpus (1)

• Fractal analysis for sets of non-differentiability of Minkowski's question mark function

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