The Assouad dimension of self-affine measures on sponges

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.