A mixed multifractal analysis for quasi Ahlfors vector-valued measures

TL;DR

This paper extends the multifractal formalism to a broad class of measures called quasi Ahlfors, developing a joint analysis for multiple measures and establishing the validity of the formalism in this context.

Contribution

It introduces a mixed multifractal analysis framework for quasi Ahlfors measures and proves the multifractal formalism holds for this class.

Findings

Developed mixed multifractal measures and dimensions.

Proved the multifractal formalism for quasi Ahlfors measures.

Established properties of these measures and dimensions.

Abstract

The multifractal formalism for measures in its original formulation is checked for special classes of measures such as doubling, self-similar, and Gibbs-like ones. Out of these classes, suitable conditions should be taken into account to prove the validity of the multifractal formalism. In the present work, a large class of measures satisfying a weak condition known as quasi Ahlfors is considered in the framework of mixed multifractal analysis. A joint multifractal analysis of finitely many quasi Ahlfors probability measures is developed. Mixed variants of multifractal generalizations of Hausdorff and packing measures, and corresponding dimensions are introduced. By applying convexity arguments, some properties of these measures and dimensions are established. Finally, an associated multifractal formalism is introduced and proved to hold for the class of quasi Ahlfors measures.