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The Assouad spectrum of random self-affine carpets

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Fraser_2020_Assouad_spectrum_ETDS_AAM.pdf (446.1Kb)
Date
15/10/2020
Author
Fraser, Jonathan
Troscheit, Sascha
Keywords
Assouad spectrum
Quasi-Assouad dimension
Random self-affine carpet
QA Mathematics
T-NDAS
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Abstract
We derive the almost sure Assouad spectrum and quasi-Assouad dimension of one-variable random self-affine Bedford–McMullen carpets. Previous work has revealed that the (related) Assouad dimension is not sufficiently sensitive to distinguish between subtle changes in the random model, since it tends to be almost surely ‘as large as possible’ (a deterministic quantity). This has been verified in conformal and non-conformal settings. In the conformal setting, the Assouad spectrum and quasi-Assouad dimension behave rather differently, tending to almost surely coincide with the upper box dimension. Here we investigate the non-conformal setting and find that the Assouad spectrum and quasi-Assouad dimension generally do not coincide with the box dimension or Assouad dimension. We provide examples highlighting the subtle differences between these notions. Our proofs combine deterministic covering techniques with suitably adapted Chernoff estimates and Borel–Cantelli-type arguments.
Citation
Fraser , J & Troscheit , S 2020 , ' The Assouad spectrum of random self-affine carpets ' , Ergodic Theory and Dynamical Systems , vol. First View . https://doi.org/10.1017/etds.2020.93
Publication
Ergodic Theory and Dynamical Systems
Status
Peer reviewed
DOI
https://doi.org/10.1017/etds.2020.93
ISSN
0143-3857
Type
Journal article
Rights
Copyright © The Author(s), 2020. Published by Cambridge University Press. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1017/etds.2020.93.
Description
Funding: JMF was financially supported by the Leverhulme Trust Research Fellowship RF-2016-500, the EPSRC Standard Grant EP/R015104/1, and the University of Waterloo. ST was financially supported by NSERC Grants 2014-03154 and 2016-03719, and the University of Waterloo.
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  • University of St Andrews Research
URI
http://hdl.handle.net/10023/23032

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