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. doi: 10.5186/aasfm.1985.1005.

[2]

J. Athreya, Logarithm laws and shrinking target properties, Proc. Indian Acad. Sci. Math. Sci., 119 (2009), 541-557. doi: 10.1007/s12044-009-0044-x.

[3]

J. Athreya, Cusp excursions on parameter spaces, J. London Math. Soc., 87 (2013), 741-765. doi: 10.1112/jlms/jds074.

[4]

J. Athreya and G. Margulis, Logarithm laws for unipotent flows. Ⅰ, J. Mod. Dyn., 3 (2009), 359-378. doi: 10.3934/jmd.2009.3.359.

[5]

J. Athreya and G. Margulis, Logarithm laws for unipotent flows. Ⅱ, J. Mod. Dyn., 11 (2017), 1-16. doi: 10.3934/jmd.2017001.

[6]

A. Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, Differential J. Geometry, 6 (1972), 543-560. doi: 10.4310/jdg/1214430642.

[7]

J. ElstrodtF. Grunewald and J. Mennicke, Vahlen's group of Clifford matrices and spingroups, Math. Z., 196 (1987), 369-390. doi: 10.1007/BF01200359.

[8]

J. ElstrodtF. Grunewald and 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. doi: 10.1007/BF01231519.

[9]

H. Garland and M. S. Raghunathan, Fundamental domains for lattices in $(\mathbb{R})$-rank 1 semi-simple groups, Ann. of Math., 92 (1970), 279-326. doi: 10.2307/1970838.

[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. doi: 10.1007/BF02342923.

[12]

D. Kelmer and A. Mohammadi, Logarithm laws for one parameter unipotent flows, Geom. Funct. Anal., 22 (2012), 756-784. doi: 10.1007/s00039-012-0181-8.

[13]

D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494. doi: 10.1007/s002220050350.

[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. doi: 10.1215/S0012-7094-86-05303-2.

[18]

D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math., 149 (1982), 215-237. doi: 10.1007/BF02392354.

[19]

T. Ton-That, Lie group representations and harmonic polynomials of a matrix variable, Trans. Amer. Math. Soc., 216 (1976), 1-46. doi: 10.1090/S0002-9947-1976-0399366-1.

[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.

show all references

References:
[1]

L. Ahlfors, On the fixed points of Möbius tranformations in $\mathbb{R}^{n}$, Annales Academiae, 10 (1985), 15-27. doi: 10.5186/aasfm.1985.1005.

[2]

J. Athreya, Logarithm laws and shrinking target properties, Proc. Indian Acad. Sci. Math. Sci., 119 (2009), 541-557. doi: 10.1007/s12044-009-0044-x.

[3]

J. Athreya, Cusp excursions on parameter spaces, J. London Math. Soc., 87 (2013), 741-765. doi: 10.1112/jlms/jds074.

[4]

J. Athreya and G. Margulis, Logarithm laws for unipotent flows. Ⅰ, J. Mod. Dyn., 3 (2009), 359-378. doi: 10.3934/jmd.2009.3.359.

[5]

J. Athreya and G. Margulis, Logarithm laws for unipotent flows. Ⅱ, J. Mod. Dyn., 11 (2017), 1-16. doi: 10.3934/jmd.2017001.

[6]

A. Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, Differential J. Geometry, 6 (1972), 543-560. doi: 10.4310/jdg/1214430642.

[7]

J. ElstrodtF. Grunewald and J. Mennicke, Vahlen's group of Clifford matrices and spingroups, Math. Z., 196 (1987), 369-390. doi: 10.1007/BF01200359.

[8]

J. ElstrodtF. Grunewald and 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. doi: 10.1007/BF01231519.

[9]

H. Garland and M. S. Raghunathan, Fundamental domains for lattices in $(\mathbb{R})$-rank 1 semi-simple groups, Ann. of Math., 92 (1970), 279-326. doi: 10.2307/1970838.

[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. doi: 10.1007/BF02342923.

[12]

D. Kelmer and A. Mohammadi, Logarithm laws for one parameter unipotent flows, Geom. Funct. Anal., 22 (2012), 756-784. doi: 10.1007/s00039-012-0181-8.

[13]

D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494. doi: 10.1007/s002220050350.

[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. doi: 10.1215/S0012-7094-86-05303-2.

[18]

D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math., 149 (1982), 215-237. doi: 10.1007/BF02392354.

[19]

T. Ton-That, Lie group representations and harmonic polynomials of a matrix variable, Trans. Amer. Math. Soc., 216 (1976), 1-46. doi: 10.1090/S0002-9947-1976-0399366-1.

[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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