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 ℝd 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 those dimensions. Moreover, there are easy-to-check conditions guaranteeing that the bounds coincide for d⩾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 ‘Barański type’ the Assouad dimension of all associated self-affine measures strictly exceeds the Assouad dimension of the carpet by some fixed δ>0 depending only on the carpet. We also provide examples of self-affine carpets of ‘Barański 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.