Geodesic
On Tori Triangulations Associated with Two-Dimensional Continued Fractions of Cubic Irrationalities
Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 2, pp. 28-37

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The notion of equivalence of multidimensional continued fractions is introduced. We consider some properties and state some conjectures related to the structure of the family of equivalence classes of two-dimensional periodic continued fractions. Our approach to the study of the family of equivalence classes of two-dimensional periodic continued fractions leads to revealing special subfamilies of continued fractions for which the triangulations of the torus (i.e., the combinatorics of their fundamental domains) are subjected to clear rules. Some of these subfamilies are studied in detail; the way to construct other similar subfamilies is indicated.
Keywords: multidimensional continued fractions, convex hulls, integer operators, cubic extensions of $\mathbb{Q}$. 
@article{FAA_2004_38_2_a2,
     author = {O. N. Karpenkov},
     title = {On {Tori} {Triangulations} {Associated} with {Two-Dimensional} {Continued} {Fractions} of {Cubic} {Irrationalities}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {28--37},
     publisher = {mathdoc},
     volume = {38},
     number = {2},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2004_38_2_a2/}
}
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O. N. Karpenkov. On Tori Triangulations Associated with Two-Dimensional Continued Fractions of Cubic Irrationalities. Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 2, pp. 28-37. http://geodesic.mathdoc.fr/item/FAA_2004_38_2_a2/