A Multifractal Formalism

Abstract

Let X be a metric space and μ a Borel probability measure on X. For q, t ∈ R and E ⊆ X write [formula]Then Hq, tμ and Pq, tμ are Borel measures − Hq, tμ is a multifractal generalization of the centered Hausdorff measure and Pq, tμ is a multifractal generalization of the packing measure. The measures Hq, tμ and Pq, tμ define, for a fixed q, in the usual way a generalized Hausdorff dimension dimqμ(E) and a generalized packing dimension Dimqμ(E) of subsets E of X. We study the functions bμ: q → dimqμ(supp μ), Bμq → Dimqμ(supp μ) and their relation to the so-called multifractal spectra functions of μ: [formula] We prove, among other things, that fμ(Fμ) is bounded from above by the Legendre transform of bμBμ) and that equality holds for graph directed self-similar measures and "cookie-cutter" measures. Finally we discuss the connection with generalized Rényi dimensions.