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

By Marc Kessebohmer, Bernd Stratmann. 2007

56 citations recorded by the source corpus.

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 Λ 0 : = { x : Q ′ ( x ) = 0 } , Λ ∞ : = { x : Q ′ ( x ) = ∞ } , and Λ ∼ : = { x : Q ′ ( x ) does not exist and Q ′ ( x ) ≠ ∞ } . The main result is that the Hausdorff dimensions of these sets are related in the following way: dim H ( ν F ) dim H ( Λ ∼ ) = dim H ( Λ ∞ ) = dim H ( L ( h top ) ) dim H ( Λ 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 ( ν 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.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/8b163cd255031db77f7c6face578eb17f6f55287

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Marc Kessebohmer
2. T. Jordan
3. Tuomas Sahlsten
4. Bernd Stratmann

In Mathematical Physics

1. Bernd Stratmann
2. Marc Kessebohmer
3. Sara Munday
4. T. Jordan
5. Tuomas Sahlsten
6. J. Barral
7. M. Mensi

In Molecular Biology

1. Marc Kessebohmer
2. Bernd Stratmann

Citations in the corpus

• Fourier transforms of Gibbs measures for the Gauss map (2013, background): “It turns out that ? has many fractal features, see for example the work by Kesseböhmer and Stratmann [15] on multifractal analysis and the references therein.”
• Strong renewal theorems and Lyapunov spectra for α-Farey and α-Lüroth systems (2010, background): “In particular, this will include a discussion of the topological dynamics of these two maps and the way in which they give rise to a family of distribution functions which are all in the spirit of the Minkowski question mark function (see [17, 23, 25]).”