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.

22 pages, 2 figures

Cited by in corpus (1)

• On the Minkowski Measure