H\"older differentiability of self-conformal devil's staircases

TL;DR

This paper investigates the H"older differentiability properties of the devil's staircase function associated with self-conformal sets, using thermodynamic formalism to determine the Hausdorff dimensions of various sets where the derivative behaves differently.

Contribution

It extends previous work by applying thermodynamic multifractal formalism to analyze the differentiability of self-conformal devil's staircases, providing new dimension results.

Findings

Calculated Hausdorff dimensions of sets with zero, infinite, or no derivative.

Extended prior results on differentiability of devil's staircases.

Used thermodynamic formalism for multifractal analysis.

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 $S^{\alpha}_{0}$, $S^{\alpha}_{\infty}$ and $S^{\alpha}$, the set of points at which this function has, respectively, H\"older derivative 0, $\infty$ or no derivative in the general sense. This extends recent work by Darst, Dekking, Falconer, Kesseb\"ohmer and Stratmann and Yao, Zhang and Li.