Mathematics – Dynamical Systems
Scientific paper
2007-06-04
Journal of Number Theory 128 (2008), 2663-2686
Mathematics
Dynamical Systems
22 pages, 2 figures
Scientific paper
10.1016/j.jnt.2007.12.010
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.
Kesseböhmer Marc
Stratmann Bernd O.
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