Hölder differentiability of self-conformal devil's staircases

Abstract

In this paper we consider the probability distribution function of a Gibbs measure supported on a self-conformal set given by an iterated function system (devil's staircase). We use thermodynamic multifractal formalism to calculate the Hausdorff dimension of the sets $S_0^{\alpha }$, $S_{\infty }^{\alpha }$ and $S^{\alpha }$, the set of points at which this function has, respectively, Hölder derivative 0, $\infty$ or no derivative in the general sense. This extends recent work by Darst, Dekking, Falconer, Kesseböhmer and Stratmann and Yao, Zhang and Li.

16 pages, 4 figures

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