On the multivariate multifractal formalism: examples and counter-examples

TL;DR

This paper explores the complexities of bivariate multifractal analysis, revealing that the natural Legendre spectrum extension does not always bound the spectrum and presenting new examples with unexpected behaviors.

Contribution

It demonstrates that the Legendre spectrum extension fails as an upper bound in bivariate cases and provides novel examples illustrating diverse multifractal behaviors.

Findings

Legendre spectrum extension does not bound the bivariate spectrum

Constructed measures with disjoint spectrum supports

Identified surprising behaviors in correlated measure pairs

Abstract

In this article, we investigate the bivariate multifractal analysis of pairs of Borel probability measures. We prove that, contrarily to what happens in the univariate case, the natural extension of the Legendre spectrum does not yield an upper bound for the bivariate multifractal spectrum. For this we build a pair of measures for which the two spectra have disjoint supports. Then we study the bivariate multifractal behavior of an archetypical pair of randomly correlated measures, which give new, surprising, behaviors, enriching the narrow class of measures for which such an analysis is achieved.