Fractal analysis for sets of non-differentiability of Minkowski's question mark function
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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 $Λ_{0}:=\{x:Q'(x)=0\}$, $Λ_{\infty}:=\{x:Q'(x)=\infty\}$, and $Λ_{\sim}:=\{x:Q'(x)$ 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. $\dim_{H}(ν_{F})<\dim_{H}(Λ_{\sim})= \dim_{H} (Λ_{\infty}) = \dim_{H} (\mathcal{L}(h_{\mathrm{top}}))<\dim_{H}(Λ_{0})=1.$
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 $\dim_{H}(ν_{F})$ 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.
22 pages, 2 figures
22 pages, 2 figures