Revisiting multifractal analysis

TL;DR

This paper presents new proofs related to the multifractal formalism, demonstrating results at points where key functions differ, and providing examples of measures with distinct Hausdorff and packing dimensions of level sets.

Contribution

It offers novel proofs of multifractal theorems and constructs examples where the multifractal spectrum functions b(q) and B(q) differ, with explicit dimension calculations.

Findings

New proofs extend the multifractal formalism to points where b(q) and B(q) differ

Constructed example measure shows b(q) and B(q) can differ significantly

Hausdorff and packing dimensions of level sets are given by Legendre transforms of b and B

Abstract

New proofs of theorems on the multifractal formalism are given. They yield results even at points q for which Olsen's functions b(q) and B(q) differ. Indeed, we provide an example of measure for which functions b and B differ and for which the Hausdorff dimension of the level sets of the local Holder exponent are given by the Legendre transform of b and their packing dimension by the Legendre transform of B.