The Assouad spectrum of random self-affine carpets
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
ISSN
0143-3857Type
Journal article
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.Collections
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