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

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 $alphain mathbb{R}$, we establish the formalism for the Hausdorff dimension of level sets of points $xinLambda$ 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 Holder regularity of various fractal functions, such as distribution functions and conjugacy maps associated with conformal iterated function systems.