Toward zeta functions and complex dimensions of multifractals

TL;DR

This paper explores the extension of zeta functions and complex dimensions from fractal strings to multifractals, aiming to develop new analytical tools for understanding their intricate structure and spectra.

Contribution

It introduces novel approaches to apply zeta functions and complex dimensions to multifractals, expanding classical fractal analysis methods.

Findings

Connected multifractal analysis with zeta functions and complex dimensions.

Proposed new perspectives for studying multifractal spectra.

Identified open questions for future research.

Abstract

Multifractals are inhomogeneous measures (or functions) which are typically described by a full spectrum of real dimensions, as opposed to a single real dimension. Results from the study of fractal strings in the analysis of their geometry, spectra and dynamics via certain zeta functions and their poles (the complex dimensions) are used in this text as a springboard to define similar tools fit for the study of multifractals such as the binomial measure. The goal of this work is to shine light on new ideas and perspectives rather than to summarize a coherent theory. Progress has been made which connects these new perspectives to and expands upon classical results, leading to a healthy variety of natural and interesting questions for further investigation and elaboration.