The Assouad dimension of self-affine measures on sponges
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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.
Fraser , J & Kolossvary , I T 2022 , ' The Assouad dimension of self-affine measures on sponges ' , Ergodic Theory and Dynamical Systems , vol. FirstView . https://doi.org/10.1017/etds.2022.64
Ergodic Theory and Dynamical Systems
Copyright © The Author(s), 2022. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
DescriptionFunding: Royal Society of Edinburgh - 70249; Leverhulme Trust - RPG-2019-034; Engineering and Physical Sciences Research Council - EP/R015104/1.
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