Hausdorff dimension, Mean quadratic variation of infinite self-similar measures

TL;DR

This paper investigates the Hausdorff dimension of infinite self-similar sets and the asymptotic behavior of the mean quadratic variation of associated measures, under weaker conditions than previously established.

Contribution

It extends the understanding of Hausdorff dimensions and quadratic variations for infinite self-similar measures beyond classical assumptions.

Findings

Hausdorff dimension of infinite self-similar sets is determined under weaker conditions.

Mean quadratic variation of the measure exhibits specific asymptotic properties.

Results generalize previous work by Riedi & Mandelbrot.

Abstract

Under weaker condition than that of Riedi & Mandelbrot, the Hausdorff (and Hausdorff-Besicovitch) dimension of infinite self-similar set K which is the invariant compact set of infinite contractive similarities {S_j(x)} satisfying open set condition is obtained. It is proved (under some additional hypotheses) that the mean quadratic variation of infinite self-similar measure is of asymptotic property.