Constructing multidimensional periodic continued fractions in the sense of Klein

Abstract

We consider the geometric generalization of ordinary continued fractions to the multidimensional case introduced by F. Klein in 1895. A multidimensional periodic continued fraction is the union of sails with some special group acting freely on these sails. This group transposes the faces. In this article, we present a method of constructing approximate fundamental domains of algebraic multidimensional continued fractions and an algorithm testing whether this domain is indeed fundamental or not. We give some polynomial estimates on the number of the operations for the algorithm. In conclusion we present an example of a fundamental domain calculation for a two-dimensional series of two-dimensional periodic continued fractions.