Representation of families of sets by measures, dimension spectra and Diophantine approximation

Abstract

A family of sets {Fd}d is said to be ‘represented by the measure μ’ if, for each d, the set Fd comprises those points at which the local dimension of μ takes some specific value (depending on d). Finding the Hausdorff dimension of these sets may then be thought of as finding the dimension spectrum, or multifractal spectrum, of μ. This situation pertains surprisingly often, with many familiar families of sets representable by measures which have simple dimension spectra. Examples are given from Diophantine approximation, Kleinian groups and hyperbolic dynamical systems.