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:
[1]

M. Babillot, On the mixing property for hyperbolic systems, Israel J. Math., 129 (2002), 61-76. doi: 10.1007/BF02773153.

[2]

A. F. Beardon, "The Geometry of Discrete Groups," Grad. Texts Math., 91, Springer-Verlag, New York, 1983.

[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.

[4]

B. Bowditch, Geometrical finiteness with variable negative curvature, Duke Math. J., 77 (1995), 229-274. doi: 10.1215/S0012-7094-95-07709-6.

[5]

M. R. Bridson and A. Haefliger, "Metric Spaces of Non-Positive Curvature," Grund. Math. Wiss., 319, Springer-Verlag, Berlin, 1999.

[6]

J. Buchmann and U. Vollmer, "Binary Quadratic Forms: An Algorithmic Approach," Alg. Comput. Math., 20, Springer, Berlin, 2007.

[7]

D. A. Buell, "Binary Quadratic Forms. Classical Theory and Modern Computations," Springer-Verlag, New York, 1989.

[8]

Y. Bugeaud, "Approximation by Algebraic Numbers," Camb. Tracts Math., 160, Cambridge Univ. Press, Cambridge, 2004.

[9]

E. Burger, A tail of two palindromes, Amer. Math. Month., 112 (2005), 311-321. doi: 10.2307/30037467.

[10]

P. Buser and H. Karcher, "Gromov's Almost Flat Manifolds," Astérisque, 81, Soc. Math. France, 1981.

[11]

H. Cohn, "A Second Course in Number Theory," John Wiley & Sons, Inc., New York-London, 1962.

[12]

F. Dal'Bo, J.-P. Otal and M. Peigné, Séries de Poincaré des groupes géométriquement finis, Israel J. Math., 118 (2000), 109-124.

[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.

[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.

[15]

A. Gorodnik and F. Paulin, Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows, preprint, arXiv:1110.5773.

[16]

O. Herrmann, Über die Verteilung der Längen geodätischer Lote in hyperbolischen Raumformen, Math. Z., 79 (1962), 323-343. doi: 10.1007/BF01193127.

[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.

[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.

[19]

H. Huber, Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen, Math. Ann., 138 (1959), 1-26. doi: 10.1007/BF01369663.

[20]

S. Katok, "Fuchsian Groups," Chicago Lectures in Mathematics, Univ. Chicago Press, Chicago, IL, 1992.

[21]

E. Landau, "Elementary Number Theory," Translated by J. E. Goodman, Chelsea Pub. Co., New York, 1958.

[22]

Y. Long, Criterion for $\SL_2(\mathbbZ$)-matrix to be conjugate to its inverse, Chin. Ann. Math. Ser. B, 23 (2002), 455-460.

[23]

C. Maclachlan and A. Reid, "The Arithmetic of Hyperbolic $3$-Manifolds," Grad. Texts in Math., 219, Springer-Verlag, New York, 2003.

[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.

[25]

W. Narkiewicz, Elementary and analytic theory of algebraic numbers, Second edition, Springer-Verlag, Berlin, PWN-Polish Scientific Publishers, Warsaw, 1990.

[26]

M. Newman, "Integral Matrices," Pure and Applied Mathematics, Vol. 45, Academic Press, New York-London, 1972.

[27]

H. Oh and N. Shah, Equidistribution and counting for orbits of geometrically finite hyperbolic groups, preprint, arXiv:1001.2096, to appear in J. Amer. Math. Soc.

[28]

O. Perron, "Die Lehre von den Kettenbrüchen," B. G. Teubner, 1913.

[29]

J. Parkkonen and F. Paulin, Equidistribution, counting and arithmetic applications, Oberwolfach Report, 29 (2010), 35-37.

[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.

[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.

[32]

J. Parkkonen and F. Paulin, On the arithmetic and geometry of binary Hamiltonian forms, With an appendix by Vincent Emery, preprint, Algebra & Number Theory, arXiv:1105.2290.

[33]

J. Parkkonen and F. Paulin, Skinning measure in negative curvature and equidistribution of equidistant submanifolds, preprint, arXiv:1202.6398.

[34]

J. Parkkonen and F. Paulin, Counting arcs in negative curvature, preprint, arXiv:1203.0175.

[35]

J. Parkkonen and F. Paulin, Counting common perpendicular arcs in negative curvature, in preparation.

[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.

[37]

T. Roblin, Ergodicité et équidistribution en courbure négative, Mémoire Soc. Math. France (N.S.), (2003), vi+96 pp.

[38]

P. Samuel, "Théorie Algébrique des Nombres," Hermann, Paris, 1967.

[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.

[40]

P. Sarnak, "Reciprocal Geodesics. Analytic Number Theory," Clay Math. Proc., 7, Amer. Math. Soc., Providence, RI, (2007), 217-237.

[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.

[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.

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.

[2]

A. F. Beardon, "The Geometry of Discrete Groups," Grad. Texts Math., 91, Springer-Verlag, New York, 1983.

[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.

[4]

B. Bowditch, Geometrical finiteness with variable negative curvature, Duke Math. J., 77 (1995), 229-274. doi: 10.1215/S0012-7094-95-07709-6.

[5]

M. R. Bridson and A. Haefliger, "Metric Spaces of Non-Positive Curvature," Grund. Math. Wiss., 319, Springer-Verlag, Berlin, 1999.

[6]

J. Buchmann and U. Vollmer, "Binary Quadratic Forms: An Algorithmic Approach," Alg. Comput. Math., 20, Springer, Berlin, 2007.

[7]

D. A. Buell, "Binary Quadratic Forms. Classical Theory and Modern Computations," Springer-Verlag, New York, 1989.

[8]

Y. Bugeaud, "Approximation by Algebraic Numbers," Camb. Tracts Math., 160, Cambridge Univ. Press, Cambridge, 2004.

[9]

E. Burger, A tail of two palindromes, Amer. Math. Month., 112 (2005), 311-321. doi: 10.2307/30037467.

[10]

P. Buser and H. Karcher, "Gromov's Almost Flat Manifolds," Astérisque, 81, Soc. Math. France, 1981.

[11]

H. Cohn, "A Second Course in Number Theory," John Wiley & Sons, Inc., New York-London, 1962.

[12]

F. Dal'Bo, J.-P. Otal and M. Peigné, Séries de Poincaré des groupes géométriquement finis, Israel J. Math., 118 (2000), 109-124.

[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.

[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.

[15]

A. Gorodnik and F. Paulin, Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows, preprint, arXiv:1110.5773.

[16]

O. Herrmann, Über die Verteilung der Längen geodätischer Lote in hyperbolischen Raumformen, Math. Z., 79 (1962), 323-343. doi: 10.1007/BF01193127.

[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.

[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.

[19]

H. Huber, Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen, Math. Ann., 138 (1959), 1-26. doi: 10.1007/BF01369663.

[20]

S. Katok, "Fuchsian Groups," Chicago Lectures in Mathematics, Univ. Chicago Press, Chicago, IL, 1992.

[21]

E. Landau, "Elementary Number Theory," Translated by J. E. Goodman, Chelsea Pub. Co., New York, 1958.

[22]

Y. Long, Criterion for $\SL_2(\mathbbZ$)-matrix to be conjugate to its inverse, Chin. Ann. Math. Ser. B, 23 (2002), 455-460.

[23]

C. Maclachlan and A. Reid, "The Arithmetic of Hyperbolic $3$-Manifolds," Grad. Texts in Math., 219, Springer-Verlag, New York, 2003.

[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.

[25]

W. Narkiewicz, Elementary and analytic theory of algebraic numbers, Second edition, Springer-Verlag, Berlin, PWN-Polish Scientific Publishers, Warsaw, 1990.

[26]

M. Newman, "Integral Matrices," Pure and Applied Mathematics, Vol. 45, Academic Press, New York-London, 1972.

[27]

H. Oh and N. Shah, Equidistribution and counting for orbits of geometrically finite hyperbolic groups, preprint, arXiv:1001.2096, to appear in J. Amer. Math. Soc.

[28]

O. Perron, "Die Lehre von den Kettenbrüchen," B. G. Teubner, 1913.

[29]

J. Parkkonen and F. Paulin, Equidistribution, counting and arithmetic applications, Oberwolfach Report, 29 (2010), 35-37.

[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.

[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.

[32]

J. Parkkonen and F. Paulin, On the arithmetic and geometry of binary Hamiltonian forms, With an appendix by Vincent Emery, preprint, Algebra & Number Theory, arXiv:1105.2290.

[33]

J. Parkkonen and F. Paulin, Skinning measure in negative curvature and equidistribution of equidistant submanifolds, preprint, arXiv:1202.6398.

[34]

J. Parkkonen and F. Paulin, Counting arcs in negative curvature, preprint, arXiv:1203.0175.

[35]

J. Parkkonen and F. Paulin, Counting common perpendicular arcs in negative curvature, in preparation.

[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.

[37]

T. Roblin, Ergodicité et équidistribution en courbure négative, Mémoire Soc. Math. France (N.S.), (2003), vi+96 pp.

[38]

P. Samuel, "Théorie Algébrique des Nombres," Hermann, Paris, 1967.

[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.

[40]

P. Sarnak, "Reciprocal Geodesics. Analytic Number Theory," Clay Math. Proc., 7, Amer. Math. Soc., Providence, RI, (2007), 217-237.

[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.

[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.

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