Typical Borel measures on $[0,1]d$ satisfy a multifractal formalism

TL;DR

This paper demonstrates that typical measures on the unit cube in  satisfy a multifractal formalism, with an explicitly computed linear spectrum that aligns with the Legendre transform of their $L^q$-spectrum.

Contribution

It explicitly computes the multifractal spectrum of typical measures on  and shows they satisfy a multifractal formalism in the Baire category sense.

Findings

Multifractal spectrum is linear with slope 1.

Spectrum starts at 0 and ends at dimension d.

Spectrum matches the Legendre transform of the $L^q$-spectrum.

Abstract

In this article, we prove that in the Baire category sense, measures supported by the unit cube of $\R^d$ typically satisfy a multifractal formalism. To achieve this, we compute explicitly the multifractal spectrum of such typical measures $\mu$. This spectrum appears to be linear with slope 1, starting from 0 at exponent 0, ending at dimension $d$ at exponent $d$, and it indeed coincides with the Legendre transform of the $L^q$-spectrum associated with typical measures $\mu$.