A 2-DIMENSIONAL ALGORITHM RELATED TO THE FAREY–BROCOT SEQUENCE

Moshchevitin and Vielhaber gave an interesting generalization of the Farey–Brocot sequence for dimension d ≥ 2 (see [N. Moshchevitin and M. Vielhaber, Moments for generalized Farey–Brocot partitions, Funct. Approx. Comment. Math.38 (2008), part 2, 131–157]). For dimension d = 2 they investigate two special cases called algorithm [Formula: see text] and algorithm [Formula: see text]. Algorithm [Formula: see text] is related to a proposal of Mönkemeyer and to Selmer algorithm (see [G. Panti, Multidimensional continued fractions and a Minkowski function, Monatsh. Math.154 (2008) 247–264]). However, algorithm [Formula: see text] seems to be related to a new type of 2-dimensional continued fractions. The content of this paper is first to describe such an algorithm and to give some of its ergodic properties. In the second part the dual algorithm is considered which behaves similar to the Parry–Daniels map.