Multidimensional continued fractions and a Minkowski function

Abstract

Abstract. 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.

The Minkowski question mark function: explicit series for the dyadic period function and moments

“…A by far not complete overview of the papers written about the Minkowski question mark function or closely related topics (Farey tree, enumeration of rationals, Stern's diatomic sequence, various 1-dimensional generalizations and generalizations to higher dimensions, statistics of denominators and Farey intervals, Hausdorff dimension and analytic properties) can be found in [1]. These works include [5], [6], [8], [9], [10], [12], [13] (this is the only paper where the moments of a certain singular distribution, a close relative of F (x), were considered), [11], [14], [16], [18], [20], [24], [25], [26], [27], [28], [29], [30], [31], [33]. The internet page [36] contains an up-to-date and exhaustive bibliographical list of papers related to the Minkowski question mark function.…”

The Moments of Minkowski Question Mark Function: The Dyadic Period Function

“…The paper (J. C. Lagarias, unpublished manuscript, 1991), deals with the interrelations among the additive continued fraction algorithm, the Farey tree, the Farey shift and the Minkowski question mark function. The motivation for the work [32] is a fact that the function ? (x) can be characterised as the unique homeomorphism of the real unit interval that conjugates the Farey map with the tent map.…”

A 2-Dimensional Algorithm Related to the Farey–brocot Sequence

“…Moshchevitin and Vielhaber give an interesting generalization of the Farey-Brocot sequence for dimension d ≥ 2 (see [1]). For dimension d = 2 they investigate two special cases called algorithm A and B. Algorithm B is related to a proposal of Mönkemeyer and to Selmer algorithm (see [3]). However, algorithm A seems to be related to a new type of 2-dimensional continued fractions.…”

“…We define the set Γ as the set of all points (u, v) such that ε n , ε n+1 is not one of the pairs (2,2) or (3,3) for n ≥ n (u, v). Then clearly T * Γ = (T * ) −1 Γ = Γ.…”