Multidimensional continued fractions and a Minkowski function

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 Monkemeyer continued fraction algorithm with an appropriate tent map.

17 pages, 3 figures. Revised version according to the referee's suggestions. Proof of Lemma 2.3 more detailed, other minor modifications. To appear in Monatshefte fur Mathematik