The $L^q$ spectrum of self-affine measures on sponges

TL;DR

This paper establishes a variational formula for the $L^q$ spectrum of self-affine measures on sponges, extending dimension theory and introducing a novel pressure function that unifies and generalizes previous results.

Contribution

It introduces a new pressure function and proves a variational formula for the $L^q$ spectrum of self-affine measures on sponges in arbitrary dimensions, extending existing theories.

Findings

Derived a variational formula for the $L^q$ spectrum.

Determined Frostman and box dimensions of these measures.

Unified and extended previous results to higher dimensions.

Abstract

In this paper a sponge in $\mathbb{R}^d$ is the attractor of an iterated function system consisting of finitely many strictly contracting affine maps whose linear part is a diagonal matrix. A suitable separation condition is introduced under which a variational formula is proved for the $L^q$ spectrum of any self-affine measure defined on a sponge for all $q\in\mathbb{R}$. Apart from some special cases, even the existence of their box dimension was not proved before. Under certain conditions the formula has a closed form which in general is an upper bound. The Frostman and box dimension of these measures is also determined. The approach unifies several existing results and extends them to arbitrary dimensions. The key ingredient is the introduction of a novel pressure function which aims to capture the growth rate of box counting quantities on sponges. We show that this pressure satisfies a variational principle which resembles the Ledrappier--Young formula for Hausdorff dimension.