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

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 ${\Lambda }_0:=\{x:Q^{\prime }(x)=0\}$, ${\Lambda }_{\infty }:=\{x:Q^{\prime }(x)=\infty \}$, and ${\Lambda }_{\sim }:=\{x:Q^{\prime }(x)$ does not exist and $Q^{\prime }(x)\ne \infty \}.$ The main result is that the Hausdorff dimensions of these sets are related in the following way. ${\dim }_H({\nu }_F)<{\dim }_H({\Lambda }_{\sim })={\dim }_H({\Lambda }_{\infty })={\dim }_H(\mathcal{L}(h_{\mathrm{top}}))<{\dim }_H({\Lambda }_0)=1.$ Here, $\mathcal{L}(h_{\mathrm{top}})$ refers to the level set of the Stern-Brocot multifractal decomposition at the topological entropy $h_{\mathrm{top}}=\log 2$ of the Farey map $F,$ and ${\dim }_H({\nu }_F)$ 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.

arXiv comment22 pages, 2 figures

References in corpus (1)

• A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates

Cited by in corpus (18)

• Strong renewal theorems and Lyapunov spectra for $\alpha$-Farey and $\alpha$-Lüroth systems
• The Minkowski question mark function: explicit series for the dyadic period function and moments
• The moments of Minkowski question mark function: the dyadic period function
• Hölder-differentiability of Gibbs distribution functions
• Generating and zeta functions, structure, spectral and analytic properties of the moments of Minkowski question mark function
• On Minkowski type question mark functions associated with even or odd continued fractions
• Pointwise Hölder Exponents of the Complex Analogues of the Takagi Function in Random Complex Dynamics
• Jimm, a Fundamental Involution
• A dichotomy between uniform distributions of the Stern-Brocot and the Farey sequence
• Pointwise perturbations of countable Markov maps
• On the derivative of the Minkowski question-mark function
• On the Minkowski Measure
• Semi-regular continued fractions and an exact formula for the moments of the Minkowski question mark function
• Multifractal analysis of the convergence exponent in continued fractions
• The Sullivan dictionary and Bowen-Series maps
• Some results for conjugate equations
• Errata and Addenda to Mathematical Constants
• Spectral gap property for random dynamics on the real line and multifractal analysis of generalised Takagi functions