Fast Dimension Spectrum for a Potential with a Logarithmic Singularity

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We regard the classic Thue–Morse diffraction measure as an equilibrium measure for a potential function with a logarithmic singularity over the doubling map. Our focus is on unusually fast scaling of the Birkhoff sums (superlinear) and of the local measure decay (superpolynomial). For several scaling functions, we show that points with this behavior are abundant in the sense of full Hausdorff dimension. At the fastest possible scaling, the corresponding rates reveal several remarkable phenomena. There is a gap between level sets for dyadic rationals and non-dyadic points, and beyond dyadic rationals, non-zero accumulation points occur only within intervals of positive length. The dependence between the smallest and the largest accumulation point also manifests itself in a non-trivial joint dimension spectrum.

Original languageEnglish
Article number40
JournalJournal of Statistical Physics
Issue number3
Publication statusPublished - 2024 Mar

Subject classification (UKÄ)

  • Mathematical Analysis

Free keywords

  • 37C45
  • 37D35
  • g-measure
  • Multifractal analysis
  • Unbounded potential


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