Multifractal Formalism for generalised local dimension spectra of Gibbs measures on the real line

TL;DR

This paper refines the multifractal formalism for Gibbs measures on the real line, enabling detailed analysis of local dimensions and regularity of fractal functions related to conformal iterated function systems.

Contribution

It establishes a refined multifractal formalism for local dimensions of Gibbs measures, extending analysis to the Hausdorff dimension of level sets and regularity of associated fractal functions.

Findings

Formalism for Hausdorff dimension of level sets established

Analysis of Hölder regularity of distribution functions conducted

Enhanced understanding of multifractal spectra for Gibbs measures

Abstract

We refine the multifractal formalism for the local dimension of a Gibbs measure $\mu$ supported on the attractor $\Lambda$ of a conformal iterated functions system on the real line. Namely, for given $\alpha\in \mathbb{R}$, we establish the formalism for the Hausdorff dimension of level sets of points $x\in\Lambda$ for which the $\mu$-measure of a ball of radius $r_{n}$ centered at $x$ obeys a power law $r_{n}{}^{\alpha}$, for a sequence $r_{n}\rightarrow0$. This allows us to investigate the H\"older regularity of various fractal functions, such as distribution functions and conjugacy maps associated with conformal iterated function systems.