Multidimensional continued fractions and a Minkowski function

TL;DR

This paper introduces an n-dimensional generalization of the Minkowski Question Mark function, creating a homeomorphism of an n-simplex that links a multidimensional continued fraction map with a tent map, extending classical one-dimensional theory.

Contribution

It constructs the first known n-dimensional Minkowski function analogue, broadening the understanding of continued fractions and dynamical conjugacies in higher dimensions.

Findings

Established the unique homeomorphism for the n-simplex

Conjugated the multidimensional continued fraction map with a tent map

Extended classical Minkowski function properties to higher dimensions

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.