Pointwise H\"older Exponents of the Complex Analogues of the Takagi Function in Random Complex Dynamics

TL;DR

This paper analyzes the pointwise H"older regularity of complex analogues of the Takagi function in random complex dynamics, revealing multifractal spectra and chaotic behavior on certain function spaces.

Contribution

It provides a dynamical description of H"older exponents for these functions and characterizes their multifractal spectra using ergodic theory.

Findings

The spectrum of H"older exponents is determined via multifractal formalism.

The bottom of the spectrum is strictly less than 1, indicating complex regularity.

The averaged system exhibits chaos on spaces of H"older continuous functions for certain exponents.

Abstract

We investigate the H\"older 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 H\"older exponents of $T$ and $C$, which allows us to determine the spectrum of pointwise H\"older 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 $- H\"older 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 $\beta >0.$