Geodesic
Symmetries of a two-dimensional continued fraction
Izvestiya. Mathematics , Tome 85 (2021) no. 4, pp. 666-680

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We describe the symmetry group of a multidimensional continued fraction. As a multidimensional generalization of continued fractions we consider Klein polyhedra. We distinguish two types of symmetries: Dirichlet symmetries, which correspond to the multiplication by units of the respective extension of $\mathbb{Q}$, and so-called palindromic symmetries. The main result is a criterion for a two-dimensional continued fraction to have palindromic symmetries, which is analogous to the well-known criterion for the continued fraction of a quadratic irrationality to have a symmetric period.
Keywords: multidimensional continued fractions, Klein polyhedra, Dirichlet's unit theorem. 
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O. N. German; I. A. Tlyustangelov. Symmetries of a two-dimensional continued fraction. Izvestiya. Mathematics , Tome 85 (2021) no. 4, pp. 666-680. http://geodesic.mathdoc.fr/item/IM2_2021_85_4_a2/