Multifractal spectra and multifractal zeta-functions

TL;DR

This paper introduces multifractal zeta-functions that precisely characterize a broad class of multifractal spectra, including those of self-conformal measures and ergodic averages, by linking them to the abscissae of convergence.

Contribution

It establishes a novel connection between multifractal spectra and zeta-functions, providing a unified framework for analyzing various multifractal properties.

Findings

Multifractal zeta-functions accurately describe multifractal spectra.

Spectra are shown to equal the abscissae of convergence of the zeta-functions.

Applicable to self-conformal measures and ergodic Birkhoff averages.

Abstract

We introduce multifractal zeta-functions providing precise information of a very general class of multifractal spectra, including, for example, the multifractal spectra of self-conformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions. More precisely, we prove that these and more general multifractal spectra equal the abscissae of convergence of the associated zetafunctions.