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On an invariant Möbius measure and the Gauss-Kuzmin face distribution. (English. Russian original) Zbl 1245.52006
Proc. Steklov Inst. Math. 258, 74-86 (2007); translation from Tr. Mat. Inst. Steklova 258, 79-92 (2007).
Summary: We study Möbius measures of the manifold of \(n\)-dimensional continued fractions in the sense of Klein. By definition any Möbius measure is invariant under the natural action of the group of projective transformations \(\text{PGL}(n+1)\) and is an integral of some form of the maximal dimension. It turns out that all Möbius measures are proportional, and the corresponding forms are written explicitly in some special coordinates. The formulae obtained allow one to compare approximately the relative frequencies of the \(n\)-dimensional faces of given integer-affine types for \(n\)-dimensional continued fractions. In this paper we make numerical calculations of some relative frequencies in the case of \(n=2\).

MSC:
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
11A55 Continued fractions
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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[1] M. O. Avdeeva and V. A. Bykovskii, ”Solution of Arnold’s Problem on Gauss-Kuzmin Statistics,” Preprint (Dal’nauka, Vladivostok, 2002).
[2] M. O. Avdeeva, ”On the Statistics of Partial Quotients of Finite Continued Fractions,” Funkts. Anal. Prilozh. 38(2), 1–11 (2004) [Funct. Anal. Appl. 38, 79–87 (2004)]. · Zbl 1205.11089 · doi:10.4213/faa103
[3] V. I. Arnold, ”A-Graded Algebras and Continued Fractions,” Commun. Pure Appl. Math. 42, 993–1000 (1989). · Zbl 0692.16012 · doi:10.1002/cpa.3160420705
[4] V. I. Arnold, ”Preface,” in Pseudoperiodic Topology (Am. Math. Soc., Providence, RI, 1999), AMS Transl., Ser. 2, 197, pp. ix–xii.
[5] V. I. Arnold, ”Higher Dimensional Continued Fractions,” Regul. Chaotic Dyn. 3(3), 10–17 (1998). · Zbl 1044.11596 · doi:10.1070/rd1998v003n03ABEH000076
[6] Arnold’s Problems (Phasis, Moscow, 2000; Springer, Berlin, 2004).
[7] V. I. Arnold, Continued Fractions (Moscow Cent. Contin. Math. Educ., Moscow, 2002) [in Russian].
[8] K. Briggs, ”Klein Polyhedra,” http://keithbriggs.info/klein-polyhedra.html
[9] A. D. Bryuno and V. I. Parusnikov, ”Klein Polyhedrals for Two Cubic Davenport Forms,” Mat. Zametki 56(4), 9–27 (1994) [Math. Notes 56, 994–1007 (1994)]. · Zbl 0844.11047
[10] E. Wirsing, ”On the Theorem of Gauss-Kusmin-Lévy and a Frobenius-Type Theorem for Function Spaces,” Acta Arith. 24, 507–528 (1974). · Zbl 0283.10032 · doi:10.4064/aa-24-5-507-528
[11] C. F. Gauss, Recherches arithmétiques (Blanchard, Paris, 1807).
[12] O. N. German, ”Sails and Hilbert Bases,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 239, 98–105 (2002) [Proc. Steklov Inst. Math. 239, 88–95 (2002)]. · Zbl 1068.52021
[13] O. N. Karpenkov, ”On Tori Triangulations Associated with Two-Dimensional Continued Fractions of Cubic Irrationalities,” Funkts. Anal. Prilozh. 38(2), 28–37 (2004) [Funct. Anal. Appl. 38, 102–110 (2004)]. · Zbl 1125.11042 · doi:10.4213/faa105
[14] O. N. Karpenkov, ”Two-Dimensional Continued Fractions of Hyperbolic Integer Matrices with Small Norm,” Usp. Mat. Nauk 59(5), 149–150 (2004) [Russ. Math. Surv. 59, 959–960 (2004)]. · Zbl 1160.11336 · doi:10.4213/rm778
[15] O. Karpenkov, ”On Some New Approach to Constructing Periodic Continued Fractions,” Preprint No. 12 (Lab. Math. Discr. CNRS, Luminy, 2004), http://iml.univ-mrs.fr/editions/preprint2004/files/karpenkov.pdf
[16] O. N. Karpenkov, ”Möbius Energy of Graphs,” Mat. Zametki 79(1), 146–149 (2006) [Math. Notes 79, 134–138 (2006)]. · Zbl 1141.57302 · doi:10.4213/mzm2683
[17] O. Karpenkov, ”Completely Empty Pyramids on Integer Lattices and Two-Dimensional Faces of Multidimensional Continued Fractions,” math.NT/0510482. · Zbl 1167.11025
[18] O. N. Karpenkov, ”Classification of Three-Dimensional Multistorey Completely Empty Convex Marked Pyramids,” Usp. Mat. Nauk 60(1), 169–170 (2005) [Russ. Math. Surv. 60, 165–166 (2005)]. · doi:10.4213/rm1397
[19] O. N. Karpenkov, ”Three Examples of Three-Dimensional Continued Fractions in the Sense of Klein,” C. R., Math., Acad. Sci. Paris 343(1), 5–7 (2006). · Zbl 1102.11007 · doi:10.1016/j.crma.2006.04.023
[20] F. Klein, ”Ueber einegeometrische Auffassung der gewöhnliche Kettenbruchentwicklung,” Nachr. Ges. Wiss. Göttingen, Math.-phys. Kl. 3, 357–359 (1895).
[21] F. Klein, ”Sur une représentation géométrique de dévelopement en fraction continue ordinaire,” Nouv. Ann. Math. 15(3), 327–331 (1896).
[22] M. L. Kontsevich and Yu. M. Suhov, ”Statistics of Klein Polyhedra and Multidimensional Continued Fractions,” in Pseudoperiodic Topology (Am. Math. Soc., Providence, RI, 1999), ASM Transl., Ser. 2, 197, pp. 9–27. · Zbl 0945.11012
[23] E. I. Korkina, ”The Simplest 2-Dimensional Continued Fraction,” in Int. Geom. Colloq., Moscow, 1993. · Zbl 0901.11003
[24] E. I. Korkina, ”La périodicité des fractions continues multidimensionnelles,” C. R. Acad. Sci. Paris, Sér. 1, 319(8), 777–780 (1994). · Zbl 0836.11023
[25] E. I. Korkina, ”Two-Dimensional Continued Fractions. The Simplest Examples,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 209, 143–166 (1995) [Proc. Steklov Inst. Math. 209, 124–144 (1995)].
[26] E. I. Korkina, ”The Simplest 2-Dimensional Continued Fraction,” J. Math. Sci. 82(5), 3680–3685 (1996). · Zbl 0901.11003 · doi:10.1007/BF02362573
[27] R. O. Kuzmin, ”On a Problem of Gauss,” Dokl. Akad. Nauk A, No. 18–19, 375–380 (1928).
[28] G. Lachaud, ”Polyèdre d’Arnol’d et voile d’un cône simplicial: analogues du théorème de Lagrange,” C. R. Acad. Sci. Paris, Sér. 1, 317(8), 711–716 (1993). · Zbl 0809.52025
[29] G. Lachaud, ”Voiles et polyèdres de Klein,” Preprint No. 95-22 (Lab. Math. Discr. CNRS, Luminy, 1995).
[30] P. Lévy, ”Sur les lois de probabilité dont dépendent les quotients complets et incomplets d’une fraction continue,” Bull. Soc. Math. France 57, 178–194 (1929). · JFM 55.0916.02 · doi:10.24033/bsmf.1150
[31] J.-O. Moussafir, ”Sales and Hilbert Bases,” Funkts. Anal. Prilozh. 34(2), 43–49 (2000) [Funct. Anal. Appl. 34, 114–118 (2000)]. · Zbl 1014.52002 · doi:10.4213/faa294
[32] J.-O. Moussafir, ”Voiles et polyèdres de Klein: Géométrie, algorithmes et statistiques,” Doc. Sci. Thèse (Univ. Paris IX-Dauphine, 2000), http://www.ceremade.dauphine.fr/:_msfr/articles/these.ps.gz
[33] R. Okazaki, ”On an Effective Determination of a Shintani’s Decomposition of the Cone \(\mathbb{R}\) + n ,” J. Math. Kyoto Univ. 33(4), 1057–1070 (1993). · Zbl 0814.11055 · doi:10.1215/kjm/1250519129
[34] J. O’Hara, Energy of Knots and Conformal Geometry (World Sci., River Edge, NJ, 2003), Ser. Knots and Everything 33.
[35] V. I. Parusnikov, ”Klein Polyhedra for the Fourth Extremal Cubic Form,” Mat. Zametki 67(1), 110–128 (2000) [Math. Notes 67, 87–102 (2000)]. · Zbl 0970.11024 · doi:10.4213/mzm819
[36] H. Tsuchihashi, ”Higher Dimensional Analogues of Periodic Continued Fractions and Cusp Singularities,” Tohoku Math. J. 35(4), 607–639 (1983). · Zbl 0585.14004 · doi:10.2748/tmj/1178228955
[37] M. H. Freedman, Z.-X. He, and Z. Wang, ”Möbius Energy of Knots and Unknots,” Ann. Math., Ser. 2, 139(1), 1–50 (1994). · Zbl 0817.57011 · doi:10.2307/2946626
[38] A. Ya. Khinchin, Continued Fractions (Fizmatgiz, Moscow, 1961; Univ. Chicago Press, Chicago, 1964).
[39] C. Hermite, ”Letter to C.D.J. Jacobi,” J. Reine Angew. Math. 40, 286 (1839).
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