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The Assouad dimension of self-affine measures on sponges

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Fraser_2022_ETDS_Assouad_Dimension_CC.pdf (459.6Kb)
Date
08/09/2022
Author
Fraser, Jonathan
Kolossvary, Istvan Tamas
Funder
The Leverhulme Trust
EPSRC
Grant ID
RPG-2019-034
EP/R015104/1
Keywords
Assouad dimension
Lower dimension
Self-affine carpet
Self-affine sponge
Dimension gap
QA Mathematics
T-NDAS
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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.
Citation
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
Publication
Ergodic Theory and Dynamical Systems
Status
Peer reviewed
DOI
https://doi.org/10.1017/etds.2022.64
ISSN
0143-3857
Type
Journal article
Rights
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.
Description
Funding: Royal Society of Edinburgh - 70249; Leverhulme Trust - RPG-2019-034; Engineering and Physical Sciences Research Council - EP/R015104/1.
Collections
  • University of St Andrews Research
URI
http://hdl.handle.net/10023/25993

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