Dixmier traces and coarse multifractal analysis

TL;DR

This paper explores how multifractal properties of measures on fractals can be characterized using spectral triples and Dixmier traces, linking noncommutative geometry with multifractal analysis.

Contribution

It introduces a novel approach to express multifractal properties via spectral triples and Dixmier traces, especially for self-similar measures.

Findings

Multifractal properties can be represented through spectral triples.

Dixmier traces relate to integrals over fractal sets.

For self-similar measures, a noncommutative integral matches an auxiliary multifractal measure.

Abstract

We show how multifractal properties of a measure supported by a fractal F contained in [0,1] may be expressed in terms of complementary intervals of F and thus in terms of spectral triples and the Dixmier trace of certain operators. For self-similar measures this leads to a noncommutative integral over $F$ equivalent to integration with respect to an auxilary multifractal measure.