Partition zeta functions, multifractal spectra, and tapestries of complex dimensions

TL;DR

This paper introduces a new multifractal spectrum based on partition zeta functions derived from measures and partitions, linking it to Hausdorff spectra and complex dimensions, especially for self-similar and atomic measures.

Contribution

It defines a novel multifractal spectrum using partition zeta functions and explores its connection to complex dimensions and Hausdorff spectra for specific measures.

Findings

Partition zeta functions characterize multifractal spectra.

The spectrum's concave envelope matches the Hausdorff multifractal spectrum.

Extension to complex dimensions for atomic measures.

Abstract

For a Borel measure and a sequence of partitions on the unit interval, we define a multifractal spectrum based on coarse Holder regularity. Specifically, the coarse Holder regularity values attained by a given measure and with respect to a sequence of partitions generate a sequence of lengths (or rather, scales) which in turn define certain Dirichlet series, called the partition zeta functions. The abscissae of convergence of these functions define a multifractal spectrum whose concave envelope is the (geometric) Hausdorff multifractal spectrum which follows from a certain type of Moran construction. We discuss at some length the important special case of self-similar measures associated with weighted iterated function systems and, in particular, certain multinomial measures. Moreover, our multifractal spectrum is shown to extend to a tapestry of complex dimensions for two specific atomic measures.