Hausdorff Dimension and Uniformity Factor of Strange Attractors

Abstract

The Hausdorff dimension ${D}_{0}$ of a strange attractor is argued to be the fixed point of a recursive relation, defined in terms of a suitable average of the smallest distances ${\ensuremath{\delta}}_{i}$ between points on the attractor. A fast numerical algorithm is developed to compute ${D}_{0}$. The spread $\ensuremath{\lambda}$ in the convergence rates towards zero of the distances ${\ensuremath{\delta}}_{i}$ (uniformity factor) as well as the stability of the fixed point are discussed in terms of the entropy of the ${\ensuremath{\delta}}_{i}$ distribution.