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A Survey of Some Arithmetic Applications of Ergodic Theory in Negative Curvature

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Ergodic Theory and Negative Curvature

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2164))

Abstract

This paper is a survey of some arithmetic applications of techniques in the geometry and ergodic theory of negatively curved Riemannian manifolds, focusing on the joint works of the authors. We describe Diophantine approximation results of real numbers by quadratic irrational ones, and we discuss various results on the equidistribution in \(\mathbb{R}\), \(\mathbb{C}\) and in the Heisenberg groups of arithmetically defined points. We explain how these results are consequences of equidistribution and counting properties of common perpendiculars between locally convex subsets in negatively curved orbifolds, proven using dynamical and ergodic properties of their geodesic flows. This exposition is based on lectures at the conference “Chaire Jean Morlet: Géométrie et systèmes dynamiques”, at the CIRM, Luminy, 2014. We thank B. Hasselblatt for his strong encouragements to write this survey.

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Notes

  1. 1.

    Actually, Poincaré in [Poi] was using another Hermitian form with signature (1, 2).

  2. 2.

    Though many references, including [Gol], attribute the notion of chains to E. Cartan, he himself attributes them to von Staudt in [Car, footnote 3].

References

  1. A. Arbieto, C. Matheus, C. Moreira, The Remarkable Effectiveness of Ergodic Theory in Number Theory. Part I. Green-Tao theorem by A. Arbieto, C. Matheus, C. Moreira; Part II. Elkies-McMullen theorem by C. Matheus. Ensaios Matematicos, vol. 17 (Sociedade Brasileira de Matemática, 2009), pp. 1–104

    Google Scholar 

  2. J. Athreya, Logarithm laws and shrinking target properties. Proc. Indian Acad. Aci. 119, 541–557 (2009)

    MathSciNet  MATH  Google Scholar 

  3. M. Babillot, On the mixing property for hyperbolic systems. Israel J. Math. 129, 61–76 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Badziahin, V. Beresnevich, S. Velani, Inhomogeneous theory of dual Diophantine approximation on manifolds. Adv. Math. 232, 1–35 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Basmajian, The orthogonal spectrum of a hyperbolic manifold. Am. Math. J. 115, 1139–1159 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. K. Belabas, S. Hersonsky, F. Paulin, Counting horoballs and rational geodesics. Bull. Lond. Math. Soc. 33, 606–612 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Y. Benoist, J.-F. Quint, Random walks on finite volume homogeneous spaces. Invent. Math. 187, 37–59 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Y. Benoist, J.-F. Quint, Stationary measures and invariant subsets of homogeneous spaces II. J. Amer. Math. Soc. 26, 659–734 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. V.I. Bernik, M.M. Dodson, Metric Diophantine Approximation on Manifolds. Cambridge Tracts in mathematics, vol. 137 (Cambridge University Press, 1999)

    Google Scholar 

  10. M. Björklund, A. Fish, Equidistribution of dilations of polynomial curves in nilmanifolds. Proc. Amer. Math. Soc. 137, 2111–2123 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Bowen, Periodic orbits for hyperbolic flows. Amer. J. Math. 94, 1–30 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  12. E. Breuillard, Local limit theorems and equidistribution of random walks on the Heisenberg group. Geom. Funct. Anal. 15, 35–82 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Bridgeman, J. Kahn, Hyperbolic volume of manifolds with geodesic boundary and orthospectra. Geom. Funct. Anal. 20, 1210–1230 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. M.R. Bridson, A. Haefliger, Metric Spaces of Non-positive Curvature. Grundlehren Der Mathematichen Wissenschaften, vol. 319 (Springer, 1999)

    Google Scholar 

  15. A. Broise-Alamichel, J. Parkkonen, F. Paulin, Equidistribution and counting under equilibrium states in negatively curved spaces and graphs of groups. Applications to non-Archimedean Diophantine approximation. Book preprint (318 p.) [arXiv:1612.06717]

    Google Scholar 

  16. Y. Bugeaud, Approximation by Algebraic Numbers. Cambridge Tracts in Mathematics, vol. 160 (Cambridge University Press, 2004)

    Google Scholar 

  17. Y. Bugeaud, Distribution Modulo One and Diophantine Approximation. Cambridge Tracts in Mathematics, vol. 193 (Cambridge University Press, 2012)

    Google Scholar 

  18. Y. Bugeaud, On the quadratic Lagrange spectrum. Math. Z. 276, 985–999 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. É. Cartan, Sur le groupe de la géométrie hypersphérique. Comment. Math. Helv. 4, 158–171 (1932)

    Article  MathSciNet  MATH  Google Scholar 

  20. L. Clozel, Démonstration de la conjecture τ. Invent. Math. 151, 297–328 (2003)

    Google Scholar 

  21. S. Cosentino, Equidistribution of parabolic fixed points in the limit set of Kleinian groups. Ergod. Theory Dyn. Syst. 19, 1437–1484 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. T. Cusick, M. Flahive, The Markoff and Lagrange spectra. Mathematical Surveys and Monographs, vol. 30 (American Mathematical Society, 1989)

    Google Scholar 

  23. J. Cygan, Wiener’s test for Brownian motion on the Heisenberg group. Coll. Math. 39, 367–373 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  24. F. Dal’Bo, Remarques sur le spectre des longueurs d’une surface et comptage. Bol. Soc. Bras. Math. 30, 199–221 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. F. Dal’Bo, Topologie du feuilletage fortement stable. Ann. Inst. Fourier 50, 981–993 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. F. Dal’Bo, J.-P. Otal, M. Peigné, Séries de Poincaré des groupes géométriquement finis. Isr. J. Math. 118, 109–124 (2000)

    Article  MATH  Google Scholar 

  27. M. Einsiedler, T. Ward, Ergodic Theory with a View Towards Number Theory. Graduate Texts in Mathematics, vol. 259 (Springer, 2011)

    Google Scholar 

  28. J. Elstrodt, F. Grunewald, J. Mennicke, Groups Acting on Hyperbolic Space: Harmonic Analysis and Number Theory. Springer Monographs in Mathematics (Springer, 1998)

    Google Scholar 

  29. A. Eskin, C. McMullen, Mixing, counting, and equidistribution in Lie groups. Duke Math. J. 71, 181–209 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  30. L. Fishman, D.S. Simmons, M. Urbański, Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces. Preprint [arXiv:1301.5630], to appear in Memoirs Amer. Math. Soc

    Google Scholar 

  31. R. Garg, A. Nevo, K. Taylor, The lattice point counting problem on the Heisenberg groups. Ann. Inst. Fourier 65, 2199–2233 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Ghosh, A. Gorodnik, A. Nevo, Metric Diophantine approximation on homogeneous varieties. Compos. Math. 150, 1435–1456 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. A. Ghosh, A. Gorodnik, A. Nevo, Diophantine approximation exponents on homogeneous varieties, in Recent Trends in Ergodic Theory and Dynamical Systems, Contemporary Mathematics, vol. 631 (American Mathematical Society, Providence, 2015), pp. 181–200

    MATH  Google Scholar 

  34. A. Ghosh, A. Gorodnik, A. Nevo, Best possible rates of distribution of dense lattice orbits in homogeneous spaces. Preprint [arXiv:1407.2824], to appear in J. Reine Angew. Math.

    Google Scholar 

  35. W.M. Goldman, Complex Hyperbolic Geometry (Oxford University Press, 1999)

    Google Scholar 

  36. A. Gorodnik, A. Nevo, The Ergodic Theory of Lattice Subgroups (Princeton University Press, 2010)

    Google Scholar 

  37. A. Gorodnik, F. Paulin, Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows. J. Mod. Dyn. 8, 25–59 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. B. Green, T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds. Ann. Math. 175, 465–540 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  39. M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics, vol. 152 (Birkhäuser, 1999)

    Google Scholar 

  40. Y. Guivarc’h, Groupes de Lie à croissance polynomiale. C.R. Acad. Sci. Paris 271, A237–A239 (1970)

    Google Scholar 

  41. U. Hamenstädt, A new description of the Bowen-Margulis measure. Ergod. Theory Dyn. Syst. 9, 455–464 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  42. G. Harcos, Equidistribution on the modular surface and L-functions, in Homogeneous Flows, Moduli Spaces and Arithmetic, ed. by M. Einsiedler et al., Clay Mathematics Proceedings, vol. 10 (American Mathematical Society, 2010), pp. 377–387

    Google Scholar 

  43. J. Heinonen, Lectures on Analysis on Metric Spaces. Universitext (Springer, 2001)

    Google Scholar 

  44. O. Herrmann, Über die Verteilung der Längen geodätischer Lote in hyperbolischen Raumformen. Math. Z. 79, 323–343 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  45. S. Hersonsky, F. Paulin, On the rigidity of discrete isometry groups of negatively curved spaces. Comment. Math. Helv. 72, 349–388 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  46. S. Hersonsky, F. Paulin, On the almost sure spiraling of geodesics in negatively curved manifolds. J. Differ. Geom. 85, 271–314 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  47. S. Hersonsky, F. Paulin, Counting orbit points in coverings of negatively curved manifolds and Hausdorff dimension of cusp excursions. Ergod. Theory Dyn. Syst. 24, 803–824 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  48. H. Huber, Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen. Math. Ann. 138, 1–26 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  49. I. Kim, Counting, mixing and equidistribution of horospheres in geometrically finite rank one locally symmetric manifolds. J. reine angew. Math. 704, 85–133 (2015)

    MathSciNet  MATH  Google Scholar 

  50. D. Kleinbock, Some applications of homogeneous dynamics to number theory, in Smooth Ergodic Theory and its Applications (Seattle, 1999). Proceedings of Symposia in Pure Mathematics, vol. 69 (American Mathematical Society, 2001), pp. 639–660

    Google Scholar 

  51. D. Kleinbock, Ergodic theory on homogeneous spaces and metric number theory, in Encyclopedia of Complexity and Systems Science, ed. by R.A. Meyers (Springer, 2009), pp. 3029–3040

    Google Scholar 

  52. D. Kleinbock, G. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinai’s Moscow Seminar on Dynamical Systems. American Mathematical Society Translations Series, vol. 171 (American Mathematical Society, 1996), pp. 141–172

    Google Scholar 

  53. D. Kleinbock, G. Margulis, Logarithm laws for flows on homogeneous spaces. Invent. Math. 138, 451–494 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  54. A. Kontorovich, The hyperbolic lattice point count in infinite volume with applications to sieves. Duke Math. J. 149, 1–36 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  55. A. Kontorovich, H. Oh, Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds. J. Am. Math. Soc. 24, 603–648 (2011)

    Google Scholar 

  56. A. Korányi, Geometric properties of Heisenberg-type groups. Adv. Math. 56, 28–38 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  57. F. Ledrappier, A renewal theorem for the distance in negative curvature, in Stochastic Analysis (Ithaca, 1993). Proceedings of Symposia in Pure Mathematics, vol. 57 (American Mathematical Society, 1995), pp. 351–360

    Google Scholar 

  58. X. Lin, Quadratic Lagrange spectrum: I. Preprint (2017)

    Google Scholar 

  59. E. Lindenstrauss, Some examples how to use measure classification in number theory, in Equidistribution in Number Theory, an Introduction. NATO Science Series, II: Mathematics, Physics and Chemistry, vol. 237 (Springer, 2007), pp. 261–303

    Google Scholar 

  60. G. Margulis, Applications of ergodic theory for the investigation of manifolds of negative curvature. Funct. Anal. Appl. 3, 335–336 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  61. G. Margulis, Certain measures that are connected with U-flows on compact manifolds. Funct. Anal. Appl. 4, 55–67 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  62. G. Margulis, On Some Aspects of the Theory of Anosov Systems. Springer Monographs in Mathematics (Springer, 2004)

    Google Scholar 

  63. K. Martin, M. McKee, E. Wambach, A relative trace formula for a compact Riemann surface. Int. J. Number Theory 7, 389–429 (2011); see webpage of first author for an erratum

    Google Scholar 

  64. A. Mohammadi, H. Oh, Matrix coefficients, counting and primes for orbits of geometrically finite groups. J. Euro. Math. Soc. 17, 837–897 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  65. H. Oh, N. Shah, Counting visible circles on the sphere and Kleinian groups, in Geometry, Topology and Dynamics in Negative Curvature, ed. by C.S. Aravinda, T. Farrell,J.-F. Lafont (ICM 2010 satellite conference, Bangalore). London Mathematical Society Lecture Note, vol. 425 (Cambridge University Press, 2016)

    Google Scholar 

  66. H. Oh, N. Shah, Equidistribution and counting for orbits of geometrically finite hyperbolic groups. J. Amer. Math. Soc. 26, 511–562 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  67. H. Oh, N. Shah, The asymptotic distribution of circles in the orbits of Kleinian groups. Invent. Math. 187, 1–35 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  68. J.-P. Otal, M. Peigné, Principe variationnel et groupes kleiniens. Duke Math. J. 125, 15–44 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  69. J. Parkkonen, F. Paulin, Spiraling spectra of geodesic lines in negatively curved manifolds. Math. Z. 268, 101–142 (2011); Erratum: Math. Z. 276, 1215–1216 (2014)

    Google Scholar 

  70. J. Parkkonen, F. Paulin, Équidistribution, comptage et approximation par irrationnels quadratiques. J. Mod. Dyn. 6, 1–40 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  71. J. Parkkonen, F. Paulin, Skinning measure in negative curvature and equidistribution of equidistant submanifolds. Ergod. Theory Dyn. Syst. 34, 1310–1342 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  72. J. Parkkonen, F. Paulin, On the arithmetic of crossratios and generalised Mertens’ formulas. Ann. Fac. Scien. Toulouse 23, 967–1022 (2014); Numéro Spécial “Aux croisements de la géométrie hyperbolique et de l’arithmétique”, F. Dal’Bo, C. Lecuire eds

    Google Scholar 

  73. J. Parkkonen, F. Paulin, Counting arcs in negative curvature, in Geometry, Topology and Dynamics in Negative Curvature, ed. by C.S. Aravinda, T. Farrell, J.-F. Lafont (ICM 2010 satellite conference, Bangalore). London Mathematical Society Lecture Note, vol. 425 (Cambridge University Press, 2016)

    Google Scholar 

  74. J. Parkkonen, F. Paulin, Counting common perpendicular arcs in negative curvature. Ergod. Theory Dyn. Syst. 37, 900–938 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  75. J. Parkkonen, F. Paulin, Counting and equidistribution in Heisenberg groups. Math. Ann. 367, 81–119 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  76. F. Paulin, M. Pollicott, B. Schapira, Equilibrium States in Negative Curvature. Astérisque, vol. 373 (Société Mathématique de France, 2015)

    Google Scholar 

  77. T. Pejkovic, Quadratic Lagrange spectrum. Math. Z. 283, 861–869 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  78. H. Poincaré, Les fonctions analytiques de deux variables et la représentation conforme. Rend. Circ. Mat. Palermo 23, 207–212 (1907)

    Article  MATH  Google Scholar 

  79. M. Pollicott, The Schottky-Klein prime function and counting functions for Fenchel double crosses (2011, preprint)

    Google Scholar 

  80. T. Roblin, Sur la fonction orbitale des groupes discrets en courbure négative. Ann. Inst. Fourier 52, 145–151 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  81. T. Roblin, Ergodicité et équidistribution en courbure négative. Mémoires de la Société Mathématique de France, vol. 95 (Société Mathématique de France, 2003)

    Google Scholar 

  82. P. Sarnak, Reciprocal geodesics. Clay Math. Proc. 7, 217–237 (2007)

    MathSciNet  MATH  Google Scholar 

  83. C. Series, The geometry of Markoff numbers. Math. Intell. 7, 20–29 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  84. J.-P. Serre, Lectures onN X (p). Chapman and Hall/CRC Research Notes in Mathematics, vol. 11 (CRC Press, 2012)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, 40014, Jyväskylä, Finland

    Jouni Parkkonen

  2. Laboratoire de Mathématique d’Orsay, UMR 8628 Université Paris-Sudet CNRS, Université Paris-Saclay, 91405, Orsay Cedex, France

    Frédéric Paulin

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  1. Jouni Parkkonen
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  1. Department of Mathematics, Tufts University, Medford, Massachusetts, USA

    Boris Hasselblatt

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Parkkonen, J., Paulin, F. (2017). A Survey of Some Arithmetic Applications of Ergodic Theory in Negative Curvature. In: Hasselblatt, B. (eds) Ergodic Theory and Negative Curvature. Lecture Notes in Mathematics, vol 2164. Springer, Cham. https://doi.org/10.1007/978-3-319-43059-1_7

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