Upper and lower bounds on the Renyi dimensions and the uniformity of multifractals

Abstract

We prove that for arbitrary probability measures the generalized Renyi dimensions D( q) satisfy the inequality D( q)( q − 1)/ q ≤ D( q')( q' − 1)/ q' ≤ D( q)( q' − 1)/ q'; q' > q, q, q' ∉ [0,1], sign q' = sign q. I n particular, 1 2 D(2) ≤ D(∞) ≤ D(2) and 1 2 D(-∞)≤D(-1)≤D(-∞) . The upper bound corresponds to uniform measures whereas the lower bound describes non-uniform measures such as measures with a power law singularity. In order to characterize intermediate cases we introduce a quantity that measures the uniformity of multifractals. We discuss the relation to the ⨍(α) spectrum and apply the formalism to several examples such as power law measures, the Feigenbaum attractor and the attractor and mode locking structure of the critical circle map.