# American Institute of Mathematical Sciences

January  2012, 6(1): 1-40. doi: 10.3934/jmd.2012.6.1

## Équidistribution, comptage et approximation par irrationnels quadratiques

 1 Department of Mathematics and Statistics, P. O. Box 35, 40014 University of Jyväskylä, Finland 2 DMA, UMR 8553 CNRS, Ecole Normale Supérieure, 45 rue d’Ulm, 75230 PARIS Cedex 05, France

Received  April 2011 Published  May 2012

Soit $M$ une variété hyperbolique de volume fini, nous montrons que les hypersurfaces équidistantes à une sous-variété $C$ de volume fini totalement géodésique s'équidistribuent dans $M$. Nous donnons une asymptotique précise du nombre de segments géodésiques de longueur au plus $t$, perpendiculaires communs à $C$ et au bord d'un voisinage cuspidal de $M$. Nous en déduisons des résultats sur le comptage d'irrationnels quadratiques sur $\mathbb{Q}$ ou sur une extension quadratique imaginaire de $\mathbb{Q}$, dans des orbites données des sous-groupes de congruence des groupes modulaires.

Let $M$ be a finite volume hyperbolic manifold. We show the equidistribution in $M$ of the equidistant hypersurfaces to a finite volume totally geodesic submanifold $C$. We prove a precise asymptotic formula on the number of geodesic arcs of lengths at most $t$, that are perpendicular to $C$ and to the boundary of a cuspidal neighbourhood of $M$. We deduce from it counting results of quadratic irrationals over $\mathbb{Q}$ or over imaginary quadratic extensions of $\mathbb{Q}$, in given orbits of congruence subgroups of the modular groups.
Citation: Jouni Parkkonen, Frédéric Paulin. Équidistribution, comptage et approximation par irrationnels quadratiques. Journal of Modern Dynamics, 2012, 6 (1) : 1-40. doi: 10.3934/jmd.2012.6.1
##### References:
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McMullen, Mixing, counting, and equidistribution in Lie groups, Duke Math. J., 71 (1993), 181-209. doi: 10.1215/S0012-7094-93-07108-6.  Google Scholar [15] A. Gorodnik and F. Paulin, Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows,, preprint, ().   Google Scholar [16] O. Herrmann, Über die Verteilung der Längen geodätischer Lote in hyperbolischen Raumformen, Math. Z., 79 (1962), 323-343. doi: 10.1007/BF01193127.  Google Scholar [17] S. Hersonsky, Covolume estimates for discrete groups of hyperbolic isometries having parabolic elements, Michigan Math. J., 40 (1993), 467-475. doi: 10.1307/mmj/1029004832.  Google Scholar [18] S. Hersonsky and F. Paulin, Counting orbit points in coverings of negatively curved manifolds and Hausdorff dimension of cusp excursions, Erg. Theo. Dyn. Sys., 24 (2004), 803-824. doi: 10.1017/S0143385703000300.  Google Scholar [19] H. Huber, Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen, Math. Ann., 138 (1959), 1-26. doi: 10.1007/BF01369663.  Google Scholar [20] S. Katok, "Fuchsian Groups," Chicago Lectures in Mathematics, Univ. Chicago Press, Chicago, IL, 1992.  Google Scholar [21] E. Landau, "Elementary Number Theory," Translated by J. E. Goodman, Chelsea Pub. Co., New York, 1958.  Google Scholar [22] Y. Long, Criterion for $\SL_2(\mathbbZ$)-matrix to be conjugate to its inverse, Chin. Ann. Math. Ser. B, 23 (2002), 455-460.  Google Scholar [23] C. Maclachlan and A. Reid, "The Arithmetic of Hyperbolic $3$-Manifolds," Grad. Texts in Math., 219, Springer-Verlag, New York, 2003.  Google Scholar [24] G. Margulis, "On Some Aspects of the Theory of Anosov Systems," With a survey by Richard Sharp: Periodic orbits of hyperbolic flows, Translated from the Russian by Valentina Vladimirovna Szulikowska, Springer Mono. Math., Springer-Verlag, Berlin, 2004.  Google Scholar [25] W. Narkiewicz, Elementary and analytic theory of algebraic numbers, Second edition, Springer-Verlag, Berlin, PWN-Polish Scientific Publishers, Warsaw, 1990.  Google Scholar [26] M. Newman, "Integral Matrices," Pure and Applied Mathematics, Vol. 45, Academic Press, New York-London, 1972.  Google Scholar [27] H. Oh and N. Shah, Equidistribution and counting for orbits of geometrically finite hyperbolic groups,, preprint, ().   Google Scholar [28] O. Perron, "Die Lehre von den Kettenbrüchen," B. G. Teubner, 1913. Google Scholar [29] J. Parkkonen and F. Paulin, Equidistribution, counting and arithmetic applications, Oberwolfach Report, 29 (2010), 35-37. Google Scholar [30] J. Parkkonen and F. Paulin, Spiraling spectra of geodesic lines in negatively curved manifolds, Math. Z., 268 (2011), 101-142. doi: 10.1007/s00209-010-0662-0.  Google Scholar [31] J. Parkkonen and F. Paulin, On the representations of integers by indefinite binary Hermitian forms, Bull. London Math. Soc., 43 (2011), 1048-1058. doi: 10.1112/blms/bdr041.  Google Scholar [32] J. Parkkonen and F. Paulin, On the arithmetic and geometry of binary Hamiltonian forms,, With an appendix by Vincent Emery, ().   Google Scholar [33] J. Parkkonen and F. Paulin, Skinning measure in negative curvature and equidistribution of equidistant submanifolds,, preprint, ().   Google Scholar [34] J. Parkkonen and F. Paulin, Counting arcs in negative curvature,, preprint, ().   Google Scholar [35] J. Parkkonen and F. Paulin, Counting common perpendicular arcs in negative curvature,, in preparation., ().   Google Scholar [36] L. Polterovich and Z. Rudnick, Stable mixing for cat maps and quasi-morphisms of the modular group, Erg. Theo. Dyn. Sys., 24 (2004), 609-619. doi: 10.1017/S0143385703000531.  Google Scholar [37] T. Roblin, Ergodicité et équidistribution en courbure négative, Mémoire Soc. Math. France (N.S.), (2003), vi+96 pp.  Google Scholar [38] P. Samuel, "Théorie Algébrique des Nombres," Hermann, Paris, 1967.  Google Scholar [39] P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theo., 15 (1982), 229-247. doi: 10.1016/0022-314X(82)90028-2.  Google Scholar [40] P. Sarnak, "Reciprocal Geodesics. Analytic Number Theory," Clay Math. Proc., 7, Amer. Math. Soc., Providence, RI, (2007), 217-237.  Google Scholar [41] G. Shimura, "Introduction to the Arithmetic Theory of Automorphic Functions," Kanô Memorial Lectures, No. 1, Publications of the Mathematical Society of Japan, No. 11, Iwanami Shoten, Publishers, Tokyo, Princeton Univ. Press, Princeton, NJ, 1971.  Google Scholar [42] M. Waldschmidt, "Diophantine Approximation on Linear Algebraic Groups. Transcendence Properties of the Exponential Function in Several Variables," Grundlehren der Mathematischen Wissenschaften, 326, Springer-Verlag, Berlin, 2000.  Google Scholar

show all references

##### References:
 [1] M. Babillot, On the mixing property for hyperbolic systems, Israel J. Math., 129 (2002), 61-76. doi: 10.1007/BF02773153.  Google Scholar [2] A. F. Beardon, "The Geometry of Discrete Groups," Grad. Texts Math., 91, Springer-Verlag, New York, 1983.  Google Scholar [3] K. Belabas, S. Hersonsky and F. Paulin, Counting horoballs and rational geodesics, Bull. Lond. Math. Soc., 33 (2001), 606-612. doi: 10.1112/S0024609301008244.  Google Scholar [4] B. Bowditch, Geometrical finiteness with variable negative curvature, Duke Math. J., 77 (1995), 229-274. doi: 10.1215/S0012-7094-95-07709-6.  Google Scholar [5] M. R. Bridson and A. Haefliger, "Metric Spaces of Non-Positive Curvature," Grund. Math. Wiss., 319, Springer-Verlag, Berlin, 1999.  Google Scholar [6] J. Buchmann and U. Vollmer, "Binary Quadratic Forms: An Algorithmic Approach," Alg. Comput. Math., 20, Springer, Berlin, 2007.  Google Scholar [7] D. A. Buell, "Binary Quadratic Forms. Classical Theory and Modern Computations," Springer-Verlag, New York, 1989.  Google Scholar [8] Y. Bugeaud, "Approximation by Algebraic Numbers," Camb. Tracts Math., 160, Cambridge Univ. Press, Cambridge, 2004.  Google Scholar [9] E. Burger, A tail of two palindromes, Amer. Math. Month., 112 (2005), 311-321. doi: 10.2307/30037467.  Google Scholar [10] P. Buser and H. Karcher, "Gromov's Almost Flat Manifolds," Astérisque, 81, Soc. Math. France, 1981.  Google Scholar [11] H. Cohn, "A Second Course in Number Theory," John Wiley & Sons, Inc., New York-London, 1962.  Google Scholar [12] F. Dal'Bo, J.-P. Otal and M. Peigné, Séries de Poincaré des groupes géométriquement finis, Israel J. Math., 118 (2000), 109-124.  Google Scholar [13] J. Elstrodt, F. Grunewald and J. Mennicke, "Groups Acting on Hyperbolic Space. Harmonic Analysis and Number Theory," Springer Mono. Math., Springer Verlag, Berlin, 1998.  Google Scholar [14] A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups, Duke Math. J., 71 (1993), 181-209. doi: 10.1215/S0012-7094-93-07108-6.  Google Scholar [15] A. Gorodnik and F. Paulin, Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows,, preprint, ().   Google Scholar [16] O. Herrmann, Über die Verteilung der Längen geodätischer Lote in hyperbolischen Raumformen, Math. Z., 79 (1962), 323-343. doi: 10.1007/BF01193127.  Google Scholar [17] S. Hersonsky, Covolume estimates for discrete groups of hyperbolic isometries having parabolic elements, Michigan Math. J., 40 (1993), 467-475. doi: 10.1307/mmj/1029004832.  Google Scholar [18] S. Hersonsky and F. Paulin, Counting orbit points in coverings of negatively curved manifolds and Hausdorff dimension of cusp excursions, Erg. Theo. Dyn. Sys., 24 (2004), 803-824. doi: 10.1017/S0143385703000300.  Google Scholar [19] H. Huber, Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen, Math. Ann., 138 (1959), 1-26. doi: 10.1007/BF01369663.  Google Scholar [20] S. Katok, "Fuchsian Groups," Chicago Lectures in Mathematics, Univ. Chicago Press, Chicago, IL, 1992.  Google Scholar [21] E. Landau, "Elementary Number Theory," Translated by J. E. Goodman, Chelsea Pub. Co., New York, 1958.  Google Scholar [22] Y. Long, Criterion for $\SL_2(\mathbbZ$)-matrix to be conjugate to its inverse, Chin. Ann. Math. Ser. B, 23 (2002), 455-460.  Google Scholar [23] C. Maclachlan and A. Reid, "The Arithmetic of Hyperbolic $3$-Manifolds," Grad. Texts in Math., 219, Springer-Verlag, New York, 2003.  Google Scholar [24] G. Margulis, "On Some Aspects of the Theory of Anosov Systems," With a survey by Richard Sharp: Periodic orbits of hyperbolic flows, Translated from the Russian by Valentina Vladimirovna Szulikowska, Springer Mono. Math., Springer-Verlag, Berlin, 2004.  Google Scholar [25] W. Narkiewicz, Elementary and analytic theory of algebraic numbers, Second edition, Springer-Verlag, Berlin, PWN-Polish Scientific Publishers, Warsaw, 1990.  Google Scholar [26] M. Newman, "Integral Matrices," Pure and Applied Mathematics, Vol. 45, Academic Press, New York-London, 1972.  Google Scholar [27] H. Oh and N. Shah, Equidistribution and counting for orbits of geometrically finite hyperbolic groups,, preprint, ().   Google Scholar [28] O. Perron, "Die Lehre von den Kettenbrüchen," B. G. Teubner, 1913. Google Scholar [29] J. Parkkonen and F. Paulin, Equidistribution, counting and arithmetic applications, Oberwolfach Report, 29 (2010), 35-37. Google Scholar [30] J. Parkkonen and F. Paulin, Spiraling spectra of geodesic lines in negatively curved manifolds, Math. Z., 268 (2011), 101-142. doi: 10.1007/s00209-010-0662-0.  Google Scholar [31] J. Parkkonen and F. Paulin, On the representations of integers by indefinite binary Hermitian forms, Bull. London Math. Soc., 43 (2011), 1048-1058. doi: 10.1112/blms/bdr041.  Google Scholar [32] J. Parkkonen and F. Paulin, On the arithmetic and geometry of binary Hamiltonian forms,, With an appendix by Vincent Emery, ().   Google Scholar [33] J. Parkkonen and F. Paulin, Skinning measure in negative curvature and equidistribution of equidistant submanifolds,, preprint, ().   Google Scholar [34] J. Parkkonen and F. Paulin, Counting arcs in negative curvature,, preprint, ().   Google Scholar [35] J. Parkkonen and F. Paulin, Counting common perpendicular arcs in negative curvature,, in preparation., ().   Google Scholar [36] L. Polterovich and Z. Rudnick, Stable mixing for cat maps and quasi-morphisms of the modular group, Erg. Theo. Dyn. Sys., 24 (2004), 609-619. doi: 10.1017/S0143385703000531.  Google Scholar [37] T. Roblin, Ergodicité et équidistribution en courbure négative, Mémoire Soc. Math. France (N.S.), (2003), vi+96 pp.  Google Scholar [38] P. Samuel, "Théorie Algébrique des Nombres," Hermann, Paris, 1967.  Google Scholar [39] P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theo., 15 (1982), 229-247. doi: 10.1016/0022-314X(82)90028-2.  Google Scholar [40] P. Sarnak, "Reciprocal Geodesics. Analytic Number Theory," Clay Math. Proc., 7, Amer. Math. Soc., Providence, RI, (2007), 217-237.  Google Scholar [41] G. Shimura, "Introduction to the Arithmetic Theory of Automorphic Functions," Kanô Memorial Lectures, No. 1, Publications of the Mathematical Society of Japan, No. 11, Iwanami Shoten, Publishers, Tokyo, Princeton Univ. Press, Princeton, NJ, 1971.  Google Scholar [42] M. Waldschmidt, "Diophantine Approximation on Linear Algebraic Groups. Transcendence Properties of the Exponential Function in Several Variables," Grundlehren der Mathematischen Wissenschaften, 326, Springer-Verlag, Berlin, 2000.  Google Scholar

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