The Assouad dimension of self-affine measures on sponges

Jonathan Fraser, Istvan Tamas Kolossvary*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)
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Abstract

We derive upper and lower bounds for the Assouad and lower dimensions of self-affine measures in ℝd generated by diagonal matrices and satisfying suitable separation conditions. The upper and lower bounds always coincide for d=2,3 , yielding precise explicit formulae for those dimensions. Moreover, there are easy-to-check conditions guaranteeing that the bounds coincide for d⩾4 . An interesting consequence of our results is that there can be a ‘dimension gap’ for such self-affine constructions, even in the plane. That is, we show that for some self-affine carpets of ‘Barański type’ the Assouad dimension of all associated self-affine measures strictly exceeds the Assouad dimension of the carpet by some fixed δ>0 depending only on the carpet. We also provide examples of self-affine carpets of ‘Barański type’ where there is no dimension gap and in fact the Assouad dimension of the carpet is equal to the Assouad dimension of a carefully chosen self-affine measure.
Original languageEnglish
Number of pages23
JournalErgodic Theory and Dynamical Systems
VolumeFirstView
Early online date8 Sept 2022
DOIs
Publication statusE-pub ahead of print - 8 Sept 2022

Keywords

  • Assouad dimension
  • Lower dimension
  • Self-affine carpet
  • Self-affine sponge
  • Dimension gap

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