On the multivariate multifractal formalism: Examples and counter-examples

In this article, we investigate the bivariate multifractal analysis of pairs of Borel probability measures. We show that, in contrast to the univariate case, the natural extension of the Legendre spectrum does not provide an upper bound for the bivariate multifractal spectrum. To establish this, we construct a pair of measures for which the two spectra have disjoint supports. We then examine the bivariate multifractal behavior of a prototypical pair of randomly correlated measures, which exhibits new and unexpected features, thereby enriching the very limited class of measures for which such an analysis has been carried out.