A multifractal zeta function for cookie cutter sets

TL;DR

This paper introduces a new multifractal zeta function that, under certain conditions, can determine the Hausdorff multifractal spectrum of measures, extending the analytical tools for multifractal analysis.

Contribution

It proposes a novel multifractal zeta function and demonstrates its ability to recover the Hausdorff multifractal spectrum for specific measures.

Findings

The abscissa of convergence of the zeta function matches the multifractal spectrum.

The approach generalizes previous multifractal analysis methods.

Conditions under which the spectrum can be obtained are established.

Abstract

Starting with the work of Lapidus and van Frankenhuysen a number of papers have introduced zeta functions as a way of capturing multifractal information. In this paper we propose a new multifractal zeta function and show that under certain conditions the abscissa of convergence yields the Hausdorff multifractal spectrum for a class of measures.