Quantifying inhomogeneity in fractal sets

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Abstract

An inhomogeneous fractal set is one which exhibits different scaling behaviour at different points. The Assouad dimension of a set is a quantity which finds the ‘most difficult location and scale’ at which to cover the set and its difference from box dimension can be thought of as a first-level overall measure of how inhomogeneous the set is. For the next level of analysis, we develop a quantitative theory of inhomogeneity by considering the measure of the set of points around which the set exhibits a given level of inhomogeneity at a certain scale. For a set of examples, a family of ( ×m, ×n )-invariant subsets of the 2-torus, we show that this quantity satisfies a Large Deviations Principle. We compare members of this family, demonstrating how the rate function gives us a deeper understanding of their inhomogeneity.
Original languageEnglish
Pages (from-to)1313-1330
JournalNonlinearity
Volume31
Issue number4
Early online date20 Feb 2018
DOIs
Publication statusPublished - Apr 2018

Keywords

  • Large deviations
  • Assouad dimension
  • Box dimension
  • Self-affine carpet

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