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The Assouad dimension of self-affine carpets with no grid structure


Authors: Jonathan M. Fraser and Thomas Jordan
Journal: Proc. Amer. Math. Soc. 145 (2017), 4905-4918
MSC (2010): Primary 28A80, 37C45
DOI: https://doi.org/10.1090/proc/13629
Published electronically: June 16, 2017
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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
Email: jmf32@st-andrews.ac.uk

Thomas Jordan
Affiliation: School of Mathematics, The University of Bristol, Bristol, BS8 1TW, United Kingdom
Email: thomas.jordan@bristol.ac.uk

DOI: https://doi.org/10.1090/proc/13629
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
Article copyright: © Copyright 2017 American Mathematical Society

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