SOME RESULTS ON THE BEHAVIOR AND ESTIMATION OF THE FRACTAL DIMENSIONS OF DISTRIBUTIONS ON ATTRACTORS

The strong interest in recent years in analyzing chaotic dynamical systems according to their asymptotic behavior has led to various definitions of fractal dimension and corresponding methods of statistical estimation. In this paper we first provide a rigorous mathematical framework for the study of dimension, focusing on pointwise dimension sigma(x) and the generalized Renyi dimensions D(q), and give a rigorous proof of inequalities first derived by Grassberger and Procaccia and Hentschel and Procaccia. We then specialize to the problem of statistical estimation of the correlation dimension nu and information dimension sigma. It has been recognized for some time that the error estimates accompanying the usual procedures (which generally involve least squares methods and nearest neighbor calculations) grossly underestimate the true statistical error involved. In least squares analyses of nu and sigma we identify sources of error not previously discussed in the literatur and address the problem of obtaining accurate error estimates. We then develop an estimation procedure for sigma which corrects for an important bias term (the local measure density) and provides confidence intervals for sigma. The general applicability of this method is illustrated with various numerical examples.