Assouad type spectra for some fractal families

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6 Citations (Scopus)

Abstract

In a previous paper we introduced a new `dimension spectrum', motivated by the Assouad dimension, designed to give precise information about the scaling structure and homogeneity of a metric space. In this paper we compute the spectrum explicitly for a range of well-studied fractal sets, including: the self-affine carpets of Bedford and McMullen, self-similar and self-conformal sets with overlaps, Mandelbrot percolation, and Moran constructions. We find that the spectrum behaves differently for each of these models and can take on a rich variety of forms. We also consider some applications, including the provision of new bi-Lipschitz invariants and bounds on a family of `tail densities' defined for subsets of the integers.
Original languageEnglish
Pages (from-to)2005-2043
JournalIndiana University Mathematics Journal
Volume67
Issue number5
Early online date26 Oct 2018
DOIs
Publication statusPublished - 2018

Keywords

  • Assouad dimension
  • Assouad spectrum
  • Self-affine carpets
  • Self-similar sets
  • Mandelbrot percolation
  • Moran constructions

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