Multidimensional Continued Fractions and a Minkowski Function

The Minkowski Question Mark function can be characterized as the unique homeomorphism of the real unit interval that conjugates the Farey map with the tent map. We construct an n-dimensional analogue of the Minkowski function as the only homeomorphism of an n-simplex that conjugates the piecewise-fractional map associated to the Mönkemeyer continued fraction algorithm with an appropriate tent map. 1. Preliminaries The nth order Farey set Fn in the real unit interval [0, 1] is defined by recursion: one starts with F0 = {0/1, 1/1} and obtains Fn by adding to Fn−1 all the Farey sums v1 ⊕ v2 = (a1 + a2)/(b1 + b2) of two consecutive elements vi = ai/bi of Fn−1. The union of all the Fn’s is the set of all rational numbers in [0, 1]. Analogously, by starting with B0 = F0 and replacing the Farey sum with the barycentric sum v1 ⊞ v2 = (v1 + v2)/2, we obtain an increasing sequence B0 ⊂ B1 ⊂ B2 ⊂ · · · , whose union is the set of all dyadic rationals in [0, 1]. For every n ≥ 0, there exists a unique order-preserving bijection from Fn to Bn. The union of these bijections is a bijection from ⋃ n≥0 Fn to ⋃ n≥0 Bn, which extends uniquely by continuity to an order-preserving bijection Φ : [0, 1] → [0, 1]. This last map is the Minkowski Question Mark function [7], [14], [19]. Among others, Φ has the following properties: (1) it is an order-preserving homeomorphism of [0, 1]; (2) it maps bijectively the rational numbers to the dyadic rationals, and the real algebraic numbers of degree ≤ 2 to the rationals (all these sets restricted to [0, 1], of course); (3) it is singular w.r.t. the Lebesgue measure λ (i.e., there exists a measurable set A ⊆ [0, 1] such that λ(A) = 1 and λ (