Revisiting the multifractal analysis of measures

Abstract

New proofs of theorems on the multifractal formalism are given. They yield results even at points $q$ for which Olsen’s functions $b(q)$ and $B(q)$ differ. Indeed, we provide an example of a measure for which the functions $b$ and $B$ differ and for which the Hausdorff dimensions of the sets $X_\alpha$ (the level sets of the local Hölder exponent) are given by the Legendre transform of $b$ and their packing dimensions by the Legendre transform of $B$.