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

Abstract

We extend the multifractal formalism for the local dimension spectrum of a Gibbs measure μ supported on the attractor Λ of a conformal iterated functions system on the real line. Namely, for α∈R, we establish the multifractal formalism for the Hausdorff dimension of the set of x∈Λ for which the μ-measure of a ball of radius rn centred at x obeys a power law rnα, for a sequence rn→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.