The Assouad dimension of self-affine measures on sponges

TL;DR

This paper establishes bounds and explicit formulas for the Assouad and lower dimensions of self-affine measures on sponges, revealing a possible dimension gap even in planar cases and providing conditions for equality of bounds.

Contribution

It provides new bounds and explicit formulas for Assouad dimensions of self-affine measures, and identifies conditions under which these bounds coincide, including in higher dimensions.

Findings

Bounds for Assouad and lower dimensions are tight in 2D and 3D.

Existence of a dimension gap in certain self-affine carpets.

Conditions under which the bounds for dimensions coincide.

Abstract

We derive upper and lower bounds for the Assouad and lower dimensions of self-affine measures in $\mathbb{R}^d$ generated by diagonal matrices and satisfying suitable separation conditions. The upper and lower bounds always coincide for $d=2,3$ yielding precise explicit formulae for the dimensions. Moreover, there are easy to check conditions guaranteeing that the bounds coincide for $d \geq 4$. An interesting consequence of our results is that there can be a `dimension gap' for such self-affine constructions, even in the plane. That is, we show that for some self-affine carpets of `Bara\'nski type' the Assouad dimension of all associated self-affine measures strictly exceeds the Assouad dimension of the carpet by some fixed $\delta>0$ depending only on the carpet. We also provide examples of self-affine carpets of `Bara\'nski type' where there is no dimension gap and in fact the Assouad dimension of the carpet is equal to the Assouad dimension of a carefully chosen self-affine measure.