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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Assouad dimension of self-affine carpets with no grid structure
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by Jonathan M. Fraser and Thomas Jordan PDF
Proc. Amer. Math. Soc. 145 (2017), 4905-4918 Request permission

Abstract:

Previous study of the Assouad dimension of planar self-affine sets has relied heavily on the underlying IFS having a ‘grid structure’, thus allowing for the use of approximate squares. We study the Assouad dimension of a class of self-affine carpets which do not have an associated grid structure. We find that the Assouad dimension is related to the box and Assouad dimensions of the (self-similar) projection of the self-affine set onto the first coordinate and to the local dimensions of the projection of a natural Bernoulli measure onto the first coordinate. In a special case we relate the Assouad dimension of the Przytycki-Urbański sets to the lower local dimensions of Bernoulli convolutions.
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Additional Information
  • Jonathan M. Fraser
  • Affiliation: School of Mathematics and Statistics, The University of St Andrews, St Andrews, KY16 9SS, Scotland
  • MR Author ID: 946983
  • Email: jmf32@st-andrews.ac.uk
  • Thomas Jordan
  • Affiliation: School of Mathematics, The University of Bristol, Bristol, BS8 1TW, United Kingdom
  • MR Author ID: 782791
  • Email: thomas.jordan@bristol.ac.uk
  • Received by editor(s): July 14, 2016
  • Received by editor(s) in revised form: December 21, 2016
  • Published electronically: June 16, 2017
  • Additional Notes: This work began while both authors were participating in the ICERM Semester Program on Dimension and Dynamics and are grateful for the stimulating atmosphere they found there. The authors thank Péter Varjú, De-Jun Feng and Xiong Jin for helpful discussions. The first author was financially supported by a Leverhulme Trust Research Fellowship, grant number RF-2016-500.
  • Communicated by: Jeremy Tyson
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 4905-4918
  • MSC (2010): Primary 28A80, 37C45
  • DOI: https://doi.org/10.1090/proc/13629
  • MathSciNet review: 3692005