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

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 $r_n\to 0$. This allows us to investigate the Hölder regularity of various fractal functions, such as distribution functions and conjugacy maps associated with conformal iterated function systems.