The Assouad spectrum of random self-affine carpets

Jonathan Fraser, Sascha Troscheit

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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.
Original languageEnglish
Number of pages19
JournalErgodic Theory and Dynamical Systems
VolumeFirst View
Early online date15 Oct 2020
DOIs
Publication statusE-pub ahead of print - 15 Oct 2020

Keywords

  • Assouad spectrum
  • Quasi-Assouad dimension
  • Random self-affine carpet

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