H\"older-differentiability of Gibbs distribution functions

TL;DR

This paper applies thermodynamic formalism to analyze the H"older-differentiability of distribution functions of Gibbs measures on fractal sets, extending previous results and clarifying their theoretical context.

Contribution

It determines the Hausdorff dimension of non-H"older-differentiability points for Gibbs measure distribution functions, linking these properties to thermodynamic formalism.

Findings

Identified the Hausdorff dimension of non-H"older-differentiability sets.

Extended previous work by integrating thermodynamic formalism.

Demonstrated the natural fit of earlier results within thermodynamic formalism.

Abstract

In this paper we give non-trivial applications of the thermodynamic formalism to the theory of distribution functions of Gibbs measures (devil's staircases) supported on limit sets of finitely generated conformal iterated function systems in $\R$. For a large class of these Gibbs states we determine the Hausdorff dimension of the set of points at which the distribution function of these measures is not $\alpha$-H\"older-differentiable. The obtained results give significant extensions of recent work by Darst, Dekking, Falconer, Li, Morris, and Xiao. In particular, our results clearly show that the results of these authors have their natural home within thermodynamic formalism.