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

In this paper we study various fractal geometric aspects of the Minkowski question mark function $Q\mathrm{.}$ We show that the unit interval can be written as the union of the three sets ${\mathrm{\Lambda }}_0:=\{x:Q^{\mathrm{\prime }}(x)=0\}$, ${\mathrm{\Lambda }}_{\mathrm{\infty }}:=\{x:Q^{\mathrm{\prime }}(x)=\mathrm{\infty }\}$, and ${\mathrm{\Lambda }}_{\sim }:=\{x:Q^{\mathrm{\prime }}(x)$ does not exist and $Q^{\mathrm{\prime }}(x)≠\mathrm{\infty }\}\mathrm{.}$ 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{t}\mathrm{o}\mathrm{p}$}))<\dim_{H}(\Lambda_{0})=1.$Here,$$\mathcal{L}$(h_{$\mathrm{t}\mathrm{o}\mathrm{p}$})$ refers to the level set of the Stern-Brocot multifractal decomposition at the topological entropy $h_{\mathrm{t}\mathrm{o}\mathrm{p}}=\log 2$ 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 $F\mathrm{.}$ The proofs rely partially on the multifractal formalism for Stern-Brocot intervals and give non-trivial applications of this formalism.