Renyi dimensions from local expansion rates

Abstract

A general self-similarity relation is shown to exist, expressing the Renyi-dimension function in terms of local expansion rates both for flows and maps. For the particular case of the information dimension, such an implicit equation yields the well-known Kaplan-Yorke relation. Moreover, it can be explicitly solved in some interesting cases, among which are two-dimensional maps with constant Jacobian. Detailed measurements are performed for the H\'enon attractor, with a very accurate estimate of its capacity. Finally, an expansion around the information dimension allows recovery of the Grassberger-Procaccia estimates in an easy way.