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

TL;DR

This paper investigates the fractal geometry of the Minkowski question mark function, analyzing the Hausdorff dimensions of sets where its derivative exhibits different behaviors, and relates these dimensions to multifractal properties of related dynamical systems.

Contribution

It establishes the Hausdorff dimensions of sets of points with different derivative behaviors of the Minkowski question mark function and connects these to multifractal analysis of Stern-Brocot intervals.

Findings

Hausdorff dimension of the set where Q'(x)=0 is 1

Dimensions of sets where Q'(x) is infinite or does not exist are equal and relate to topological entropy

The dimension of the measure of maximal entropy is less than that of the other sets

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 $\Lambda_{0}:=\{x:Q'(x)=0\}$, $\Lambda_{\infty}:=\{x:Q'(x)=\infty\}$, and $\Lambda_{\sim}:=\{x:Q'(x)$ does not exist and $Q'(x)\not=\infty\}.$ 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 $h_{\mathrm{top}}=\log2$ 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.