Fractal analysis for sets of non-differentiability of Minkowski's question mark function

Marc Kesseboehmer, Bernd O. Stratmann

Research output: Contribution to journalArticlepeer-review

24 Citations (Scopus)

Abstract

In this paper we study various fractal geometric aspects of the Minkowski question mark function Q. We show that the unit interval can be written as the union of the three sets Lambda(0) := {x: Q'(x) = 0}, Lambda infinity := {x: Q'(x) = infinity), and Lambda similar to := {x: Q'(x) does not exist and Q'(x) not equal infinity}. The main result is that the Hausdorff dimensions of these sets are related in the following way:

dim(H)(v(F)) < dim(H)(Lambda(similar to)) = dim(H)(Lambda(infinity)) = dim(H)(l (h(top))) < dim(H)(Lambda(0)) = 1.

Here, l(h(top)) refers to the level set of the Stern-Brocot multifractal decomposition at the topological entropy h(top) = log 2 of the Farey map F, and dim(H) (v(F)) denotes the Hausdorff dimension of the measure of maximal entropy of the dynamical system associated with F. The proofs rely partially on the multifractal formalism for Stern-Brocot intervals and give non-trivial applications of this formalism. (C) 2008 Elsevier Inc. All rights reserved.

Original languageEnglish
Pages (from-to)2663-2686
Number of pages24
JournalJournal of Number Theory
Volume128
DOIs
Publication statusPublished - Sept 2008

Keywords

  • Minkowski question mark function
  • singular functions
  • Stern-Brocot spectrum
  • Farey map
  • FINITE KLEINIAN-GROUPS
  • MULTIFRACTAL ANALYSIS
  • HAUSDORFF DIMENSION
  • GROWTH-RATES
  • POINTS

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