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

We extend the multifractal formalism for the local dimension spectrum of a Gibbs measure mu supported on the attractor Lambda of a conformal iterated functions system on the real line. Namely, for alpha is an element of R, we establish the multifractal formalism for the Hausdorff dimension of the set of x is an element of Lambda for which the mu-measure of a ball of radius r(n) centred at x obeys a power law r(n)(alpha), for a sequence r(n) -> 0. This allows us to investigate the Holder regularity of various fractal functions, such as distribution functions and conjugacy maps associated with conformal iterated function systems. (C) 2020 Elsevier Inc. All rights reserved.