Rate of convergence: the packing and centered Hausdoff measures of totally disconnected self-similar sets

TL;DR

This paper analyzes the convergence rates of algorithms for computing the centered Hausdorff and packing measures of totally disconnected self-similar sets, providing empirical bounds and confirming some conjectures.

Contribution

It introduces and empirically evaluates algorithms for accurately estimating measures of self-similar sets, especially for challenging contraction ratios.

Findings

Algorithms achieve high-precision estimates, up to 14 decimal places.

Confirmed some conjectural measure values for specific self-similar sets.

Identified computational limitations near connected cases.

Abstract

In this paper we obtain the rates of convergence of the algorithms given in [13] and [14] for an automatic computation of the centered Hausdorff and packing measures of a totally disconnected self-similar set. We evaluate these rates empirically through the numerical analysis of three standard classes of self-similar sets, namely, the families of Cantor type sets in the real line and the plane and the class of Sierpinski gaskets. For these three classes and for small contraction ratios, sharp bounds for the exact values of the corresponding measures are obtained and it is shown how these bounds automatically yield estimates of the corresponding measures, accurate in some cases to as many as 14 decimal places. In particular, the algorithms accurately recover the exact values of the measures in all cases in which these values are known by geometrical arguments. Positive results, which confirm some conjectural values given in [13] and [14] for the measures, are also obtained for an intermediate range of larger contraction ratios. We give an argument showing that, for this range of contraction ratios, the problem is inherently computational in the sense that any theoretical proof, such as those mentioned above, might be impossible, so that in these cases, our method is the only available approach. For contraction ratios close to those of the connected case our computational method becomes intractably time consuming, so the computation of the exact values of the packing and centered Hausdorff measures in the general case, with the open set condition, remains a challenging problem.