# American Institue of Mathematical Sciences

2017, 11: 447-476. doi: 10.3934/jmd.2017018

## Logarithm laws for unipotent flows on hyperbolic manifolds

 Department of Mathematics, Boston College, Chestnut Hill, MA 02467-3806, USA

Received  November 14, 2016 Revised  March 29, 2017 Published  November 2017

We prove logarithm laws for unipotent flows on non-compact finite-volume hyperbolic manifolds. Our method depends on the estimate of norms of certain incomplete Eisenstein series.

Citation: Shucheng Yu. Logarithm laws for unipotent flows on hyperbolic manifolds. Journal of Modern Dynamics, 2017, 11: 447-476. doi: 10.3934/jmd.2017018
##### References:
 [1] L. Ahlfors, On the fixed points of Möbius tranformations in $\mathbb{R}^{n}$, Annales Academiae, 10 (1985), 15-27. [2] J. Athreya, Logarithm laws and shrinking target properties, Proc. Indian Acad. Sci. Math. Sci., 119 (2009), 541-557. [3] J. Athreya, Cusp excursions on parameter spaces, J. London Math. Soc., 87 (2013), 741-765. [4] J. Athreya, G. Margulis, Logarithm laws for unipotent flows. Ⅰ, J. Mod. Dyn., 3 (2009), 359-378. [5] J. Athreya, G. Margulis, Logarithm laws for unipotent flows. Ⅱ, J. Mod. Dyn., 11 (2017), 1-16. [6] A. Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, Differential J. Geometry, 6 (1972), 543-560. [7] J. Elstrodt, F. Grunewald, J. Mennicke, Vahlen's group of Clifford matrices and spingroups, Math. Z., 196 (1987), 369-390. [8] J. Elstrodt, F. Grunewald, J. Mennicke, Kloosterman sums for Clifford algebras and a lower bound for the positive eigenvalues of the Laplacian for congruence subgroups acting on hyperbolic spaces, Invent. Math., 101 (1990), 641-685. [9] H. Garland, M. S. Raghunathan, Fundamental domains for lattices in $(\mathbb{R})$-rank 1 semi-simple groups, Ann. of Math., 92 (1970), 279-326. [10] P. Garrett, Harmonic analysis on spheres, http://www.math.umn.edu/~garrett/m/mfms/notes_2013-14/09_spheres.pdf. [11] V. Gritsenko, Arithmetic of quaternions and Eisenstein Series, translation in J. Soviet Math., 52 (1990), 3056-3063. [12] D. Kelmer, A. Mohammadi, Logarithm laws for one parameter unipotent flows, Geom. Funct. Anal., 22 (2012), 756-784. [13] D. Y. Kleinbock, G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494. [14] A. W. Knapp, Lie Groups Beyond an Introduction, Second Edition, Progress in Mathematics, vol. 140, Birkhäuser, Boston, 2002. [15] R. P. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Math., SLN 544, Berlin-Heidelberg-New York, 1976. [16] J. R. Parker, Hyperbolic spaces, Jyväskylä Lectures in Mathematics 2, 2008. [17] C. D. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J., 53 (1986), 43-65. [18] D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math., 149 (1982), 215-237. [19] T. Ton-That, Lie group representations and harmonic polynomials of a matrix variable, Trans. Amer. Math. Soc., 216 (1976), 1-46. [20] G. Warner, Selberg's trace formula for non-uniform lattices: The $\mathbb{R}$-rank one case, in Studies in Algebra and Number Theory, Adv. in Math. Suppl. Stud., 6, Academic Press, New York-London, 1979, 1-142.

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##### References:
 [1] L. Ahlfors, On the fixed points of Möbius tranformations in $\mathbb{R}^{n}$, Annales Academiae, 10 (1985), 15-27. [2] J. Athreya, Logarithm laws and shrinking target properties, Proc. Indian Acad. Sci. Math. Sci., 119 (2009), 541-557. [3] J. Athreya, Cusp excursions on parameter spaces, J. London Math. Soc., 87 (2013), 741-765. [4] J. Athreya, G. Margulis, Logarithm laws for unipotent flows. Ⅰ, J. Mod. Dyn., 3 (2009), 359-378. [5] J. Athreya, G. Margulis, Logarithm laws for unipotent flows. Ⅱ, J. Mod. Dyn., 11 (2017), 1-16. [6] A. Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, Differential J. Geometry, 6 (1972), 543-560. [7] J. Elstrodt, F. Grunewald, J. Mennicke, Vahlen's group of Clifford matrices and spingroups, Math. Z., 196 (1987), 369-390. [8] J. Elstrodt, F. Grunewald, J. Mennicke, Kloosterman sums for Clifford algebras and a lower bound for the positive eigenvalues of the Laplacian for congruence subgroups acting on hyperbolic spaces, Invent. Math., 101 (1990), 641-685. [9] H. Garland, M. S. Raghunathan, Fundamental domains for lattices in $(\mathbb{R})$-rank 1 semi-simple groups, Ann. of Math., 92 (1970), 279-326. [10] P. Garrett, Harmonic analysis on spheres, http://www.math.umn.edu/~garrett/m/mfms/notes_2013-14/09_spheres.pdf. [11] V. Gritsenko, Arithmetic of quaternions and Eisenstein Series, translation in J. Soviet Math., 52 (1990), 3056-3063. [12] D. Kelmer, A. Mohammadi, Logarithm laws for one parameter unipotent flows, Geom. Funct. Anal., 22 (2012), 756-784. [13] D. Y. Kleinbock, G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494. [14] A. W. Knapp, Lie Groups Beyond an Introduction, Second Edition, Progress in Mathematics, vol. 140, Birkhäuser, Boston, 2002. [15] R. P. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Math., SLN 544, Berlin-Heidelberg-New York, 1976. [16] J. R. Parker, Hyperbolic spaces, Jyväskylä Lectures in Mathematics 2, 2008. [17] C. D. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J., 53 (1986), 43-65. [18] D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math., 149 (1982), 215-237. [19] T. Ton-That, Lie group representations and harmonic polynomials of a matrix variable, Trans. Amer. Math. Soc., 216 (1976), 1-46. [20] G. Warner, Selberg's trace formula for non-uniform lattices: The $\mathbb{R}$-rank one case, in Studies in Algebra and Number Theory, Adv. in Math. Suppl. Stud., 6, Academic Press, New York-London, 1979, 1-142.
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