The Assouad spectrum of random self-affine carpets
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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.
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
Ergodic Theory and Dynamical Systems
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.
DescriptionFunding: 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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