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 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

 Funktsional. Anal. i Prilozhen., 2004, Volume 38, Issue 2, Pages 28–37 (Mi faa105)

On Tori Triangulations Associated with Two-Dimensional Continued Fractions of Cubic Irrationalities

O. N. Karpenkov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: 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 $\mathbb{Q}$

DOI: https://doi.org/10.4213/faa105

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English version:
Functional Analysis and Its Applications, 2004, 38:2, 102–110

Bibliographic databases:

UDC: 511.9

Citation: O. N. Karpenkov, “On Tori Triangulations Associated with Two-Dimensional Continued Fractions of Cubic Irrationalities”, Funktsional. Anal. i Prilozhen., 38:2 (2004), 28–37; Funct. Anal. Appl., 38:2 (2004), 102–110

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa105
• https://doi.org/10.4213/faa105
• http://mi.mathnet.ru/eng/faa/v38/i2/p28

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. O. N. Karpenkov, “Classification of three-dimensional multistorey completely empty convex marked pyramids”, Russian Math. Surveys, 60:1 (2005), 165–166
2. Karpenkov O., “Three examples of three-dimensional continued fractions in the sense of Klein”, C. R. Math. Acad. Sci. Paris, 343:1 (2006), 5–7
3. O. N. Karpenkov, “On an Invariant Möbius Measure and the Gauss–Kuzmin Face Distribution”, Proc. Steklov Inst. Math., 258 (2007), 74–86
4. Karpenkov O.N., “Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions”, Monatsh. Math., 152:3 (2007), 217–249
5. Karpenkov O., “Elementary notions of lattice trigonometry”, Math. Scand., 102:2 (2008), 161–205
6. Karpenkov O.N., “Constructing multidimensional periodic continued fractions in the sense of Klein”, Math. Comp., 78:267 (2009), 1687–1711
7. O. N. Karpenkov, “Determination of Periods of Geometric Continued Fractions for Two-Dimensional Algebraic Hyperbolic Operators”, Math. Notes, 88:1 (2010), 28–38
8. Karpenkov O.N., Vershik A.M., “Rational approximation of maximal commutative subgroups of GL(n, R)”, J Fixed Point Theory Appl, 7:1 (2010), 241–263
9. Karpenkov O., “Continued fractions and the second Kepler law”, Manuscripta Math, 134:1–2 (2011), 157–169
10. Karpenkov O., “Multidimensional Gauss Reduction Theory for Conjugacy Classes of Sl(N, Z)”, J. Theor. Nr. Bordx., 25:1 (2013), 99–109
11. A. A. Illarionov, “Some properties of three-dimensional Klein polyhedra”, Sb. Math., 206:4 (2015), 510–539
12. A. A. Lodkin, “Parus Kleina i diofantovy priblizheniya vektora”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye i algoritmicheskie metody. XXX, Zap. nauchn. sem. POMI, 481, POMI, SPb., 2019, 63–73
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