Estimating the Hausdorff measure by recurrence

TL;DR

This paper introduces a novel technique for estimating the Hausdorff measure of fractal sets from below by analyzing the recurrence times of points returning close to their initial position.

Contribution

The paper presents a new recurrence-based method for lower bounds estimation of Hausdorff measures of fractals, advancing measurement techniques in fractal geometry.

Findings

Effective lower bound estimation of Hausdorff measure

Utilizes recurrence times of points

Applicable to fractal sets

Abstract

We show a new method of estimating the Hausdorff measure (of the proper dimension) of a fractal set from below. The method requires computing the subsequent closest return times of a point to itself.