Fractal analysis for sets of non-differentiability of Minkowski's question mark function
| dc.creator | Kesseböhmer, Marc | |
| dc.creator | Stratmann, Bernd O. | |
| dc.date | 2007-06-04 | |
| dc.date | 2007-06-20 | |
| dc.date.accessioned | 2026-07-07T10:07:17Z | |
| dc.date.available | 2026-07-07T10:07:17Z | |
| dc.description | 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. | |
| dc.description | 22 pages, 2 figures | |
| dc.identifier | https://arxiv.org/abs/0706.0453 | |
| dc.identifier | http://arxiv.org/abs/0706.0453 | |
| dc.identifier | Journal of Number Theory 128 (2008), 2663-2686 | |
| dc.identifier | doi:10.1016/j.jnt.2007.12.010 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/170580 | |
| dc.subject | Dynamical Systems | |
| dc.subject | Number Theory | |
| dc.subject | 26A30, 10K50 | |
| dc.title | Fractal analysis for sets of non-differentiability of Minkowski's question mark function | |
| dc.type | text |