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Fast Dimension Spectrum for a Potential with a Logarithmic Singularity

Gohlke, Philipp LU ; Lamprinakis, Georgios LU and Schmeling, Jörg LU (2024) In Journal of Statistical Physics 191(3).
Abstract

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... (More)

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.

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author
; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
37C45, 37D35, g-measure, Multifractal analysis, Unbounded potential
in
Journal of Statistical Physics
volume
191
issue
3
article number
40
publisher
Springer
external identifiers
  • scopus:85187898679
ISSN
0022-4715
DOI
10.1007/s10955-024-03252-5
language
English
LU publication?
yes
id
e2666523-9f2e-4e16-aad0-20fa91b69368
date added to LUP
2024-04-10 13:35:02
date last changed
2024-04-10 13:35:31
@article{e2666523-9f2e-4e16-aad0-20fa91b69368,
  abstract     = {{<p>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.</p>}},
  author       = {{Gohlke, Philipp and Lamprinakis, Georgios and Schmeling, Jörg}},
  issn         = {{0022-4715}},
  keywords     = {{37C45; 37D35; g-measure; Multifractal analysis; Unbounded potential}},
  language     = {{eng}},
  number       = {{3}},
  publisher    = {{Springer}},
  series       = {{Journal of Statistical Physics}},
  title        = {{Fast Dimension Spectrum for a Potential with a Logarithmic Singularity}},
  url          = {{http://dx.doi.org/10.1007/s10955-024-03252-5}},
  doi          = {{10.1007/s10955-024-03252-5}},
  volume       = {{191}},
  year         = {{2024}},
}