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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 |
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.
References:
[1] |
M. Babillot, On the mixing property for hyperbolic systems,, Israel J. Math., 129 (2002), 61.
doi: 10.1007/BF02773153. |
[2] |
A. F. Beardon, "The Geometry of Discrete Groups,", Grad. Texts Math., 91 (1983).
|
[3] |
K. Belabas, S. Hersonsky and F. Paulin, Counting horoballs and rational geodesics,, Bull. Lond. Math. Soc., 33 (2001), 606.
doi: 10.1112/S0024609301008244. |
[4] |
B. Bowditch, Geometrical finiteness with variable negative curvature,, Duke Math. J., 77 (1995), 229.
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 (1999).
|
[6] |
J. Buchmann and U. Vollmer, "Binary Quadratic Forms: An Algorithmic Approach,", Alg. Comput. Math., 20 (2007).
|
[7] |
D. A. Buell, "Binary Quadratic Forms. Classical Theory and Modern Computations,", Springer-Verlag, (1989).
|
[8] |
Y. Bugeaud, "Approximation by Algebraic Numbers,", Camb. Tracts Math., 160 (2004).
|
[9] |
E. Burger, A tail of two palindromes,, Amer. Math. Month., 112 (2005), 311.
doi: 10.2307/30037467. |
[10] |
P. Buser and H. Karcher, "Gromov's Almost Flat Manifolds,", Astérisque, 81 (1981).
|
[11] |
H. Cohn, "A Second Course in Number Theory,", John Wiley & Sons, (1962).
|
[12] |
F. Dal'Bo, J.-P. Otal and M. Peigné, Séries de Poincaré des groupes géométriquement finis,, Israel J. Math., 118 (2000), 109.
|
[13] |
J. Elstrodt, F. Grunewald and J. Mennicke, "Groups Acting on Hyperbolic Space. Harmonic Analysis and Number Theory,", Springer Mono. Math., (1998).
|
[14] |
A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups,, Duke Math. J., 71 (1993), 181.
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, (). Google Scholar |
[16] |
O. Herrmann, Über die Verteilung der Längen geodätischer Lote in hyperbolischen Raumformen,, Math. Z., 79 (1962), 323.
doi: 10.1007/BF01193127. |
[17] |
S. Hersonsky, Covolume estimates for discrete groups of hyperbolic isometries having parabolic elements,, Michigan Math. J., 40 (1993), 467.
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.
doi: 10.1017/S0143385703000300. |
[19] |
H. Huber, Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen,, Math. Ann., 138 (1959), 1.
doi: 10.1007/BF01369663. |
[20] |
S. Katok, "Fuchsian Groups,", Chicago Lectures in Mathematics, (1992).
|
[21] |
E. Landau, "Elementary Number Theory,", Translated by J. E. Goodman, (1958).
|
[22] |
Y. Long, Criterion for $\SL_2(\mathbbZ$)-matrix to be conjugate to its inverse,, Chin. Ann. Math. Ser. B, 23 (2002), 455.
|
[23] |
C. Maclachlan and A. Reid, "The Arithmetic of Hyperbolic $3$-Manifolds,", Grad. Texts in Math., 219 (2003).
|
[24] |
G. Margulis, "On Some Aspects of the Theory of Anosov Systems,", With a survey by Richard Sharp: \emph{Periodic orbits of hyperbolic flows}, (2004).
|
[25] |
W. Narkiewicz, Elementary and analytic theory of algebraic numbers,, Second edition, (1990).
|
[26] |
M. Newman, "Integral Matrices,", Pure and Applied Mathematics, (1972).
|
[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. Google Scholar |
[30] |
J. Parkkonen and F. Paulin, Spiraling spectra of geodesic lines in negatively curved manifolds,, Math. Z., 268 (2011), 101.
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.
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, (). 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.
doi: 10.1017/S0143385703000531. |
[37] |
T. Roblin, Ergodicité et équidistribution en courbure négative,, Mémoire Soc. Math. France (N.S.), (2003).
|
[38] |
P. Samuel, "Théorie Algébrique des Nombres,", Hermann, (1967).
|
[39] |
P. Sarnak, Class numbers of indefinite binary quadratic forms,, J. Number Theo., 15 (1982), 229.
doi: 10.1016/0022-314X(82)90028-2. |
[40] |
P. Sarnak, "Reciprocal Geodesics. Analytic Number Theory,", Clay Math. Proc., 7 (2007), 217.
|
[41] |
G. Shimura, "Introduction to the Arithmetic Theory of Automorphic Functions,", Kanô Memorial Lectures, (1971).
|
[42] |
M. Waldschmidt, "Diophantine Approximation on Linear Algebraic Groups. Transcendence Properties of the Exponential Function in Several Variables,", Grundlehren der Mathematischen Wissenschaften, 326 (2000).
|
show all references
References:
[1] |
M. Babillot, On the mixing property for hyperbolic systems,, Israel J. Math., 129 (2002), 61.
doi: 10.1007/BF02773153. |
[2] |
A. F. Beardon, "The Geometry of Discrete Groups,", Grad. Texts Math., 91 (1983).
|
[3] |
K. Belabas, S. Hersonsky and F. Paulin, Counting horoballs and rational geodesics,, Bull. Lond. Math. Soc., 33 (2001), 606.
doi: 10.1112/S0024609301008244. |
[4] |
B. Bowditch, Geometrical finiteness with variable negative curvature,, Duke Math. J., 77 (1995), 229.
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 (1999).
|
[6] |
J. Buchmann and U. Vollmer, "Binary Quadratic Forms: An Algorithmic Approach,", Alg. Comput. Math., 20 (2007).
|
[7] |
D. A. Buell, "Binary Quadratic Forms. Classical Theory and Modern Computations,", Springer-Verlag, (1989).
|
[8] |
Y. Bugeaud, "Approximation by Algebraic Numbers,", Camb. Tracts Math., 160 (2004).
|
[9] |
E. Burger, A tail of two palindromes,, Amer. Math. Month., 112 (2005), 311.
doi: 10.2307/30037467. |
[10] |
P. Buser and H. Karcher, "Gromov's Almost Flat Manifolds,", Astérisque, 81 (1981).
|
[11] |
H. Cohn, "A Second Course in Number Theory,", John Wiley & Sons, (1962).
|
[12] |
F. Dal'Bo, J.-P. Otal and M. Peigné, Séries de Poincaré des groupes géométriquement finis,, Israel J. Math., 118 (2000), 109.
|
[13] |
J. Elstrodt, F. Grunewald and J. Mennicke, "Groups Acting on Hyperbolic Space. Harmonic Analysis and Number Theory,", Springer Mono. Math., (1998).
|
[14] |
A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups,, Duke Math. J., 71 (1993), 181.
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, (). Google Scholar |
[16] |
O. Herrmann, Über die Verteilung der Längen geodätischer Lote in hyperbolischen Raumformen,, Math. Z., 79 (1962), 323.
doi: 10.1007/BF01193127. |
[17] |
S. Hersonsky, Covolume estimates for discrete groups of hyperbolic isometries having parabolic elements,, Michigan Math. J., 40 (1993), 467.
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.
doi: 10.1017/S0143385703000300. |
[19] |
H. Huber, Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen,, Math. Ann., 138 (1959), 1.
doi: 10.1007/BF01369663. |
[20] |
S. Katok, "Fuchsian Groups,", Chicago Lectures in Mathematics, (1992).
|
[21] |
E. Landau, "Elementary Number Theory,", Translated by J. E. Goodman, (1958).
|
[22] |
Y. Long, Criterion for $\SL_2(\mathbbZ$)-matrix to be conjugate to its inverse,, Chin. Ann. Math. Ser. B, 23 (2002), 455.
|
[23] |
C. Maclachlan and A. Reid, "The Arithmetic of Hyperbolic $3$-Manifolds,", Grad. Texts in Math., 219 (2003).
|
[24] |
G. Margulis, "On Some Aspects of the Theory of Anosov Systems,", With a survey by Richard Sharp: \emph{Periodic orbits of hyperbolic flows}, (2004).
|
[25] |
W. Narkiewicz, Elementary and analytic theory of algebraic numbers,, Second edition, (1990).
|
[26] |
M. Newman, "Integral Matrices,", Pure and Applied Mathematics, (1972).
|
[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. Google Scholar |
[30] |
J. Parkkonen and F. Paulin, Spiraling spectra of geodesic lines in negatively curved manifolds,, Math. Z., 268 (2011), 101.
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.
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, (). 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.
doi: 10.1017/S0143385703000531. |
[37] |
T. Roblin, Ergodicité et équidistribution en courbure négative,, Mémoire Soc. Math. France (N.S.), (2003).
|
[38] |
P. Samuel, "Théorie Algébrique des Nombres,", Hermann, (1967).
|
[39] |
P. Sarnak, Class numbers of indefinite binary quadratic forms,, J. Number Theo., 15 (1982), 229.
doi: 10.1016/0022-314X(82)90028-2. |
[40] |
P. Sarnak, "Reciprocal Geodesics. Analytic Number Theory,", Clay Math. Proc., 7 (2007), 217.
|
[41] |
G. Shimura, "Introduction to the Arithmetic Theory of Automorphic Functions,", Kanô Memorial Lectures, (1971).
|
[42] |
M. Waldschmidt, "Diophantine Approximation on Linear Algebraic Groups. Transcendence Properties of the Exponential Function in Several Variables,", Grundlehren der Mathematischen Wissenschaften, 326 (2000).
|
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