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.

[1]

Jayadev S. Athreya, Gregory A. Margulis. Logarithm laws for unipotent flows, Ⅱ. Journal of Modern Dynamics, 2017, 11: 1-16. doi: 10.3934/jmd.2017001

[2]

Jayadev S. Athreya, Gregory A. Margulis. Logarithm laws for unipotent flows, I. Journal of Modern Dynamics, 2009, 3 (3) : 359-378. doi: 10.3934/jmd.2009.3.359

[3]

J. S. Athreya, Anish Ghosh, Amritanshu Prasad. Ultrametric logarithm laws I. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 337-348. doi: 10.3934/dcdss.2009.2.337

[4]

Li-Xin Zhang. On the laws of the iterated logarithm under sub-linear expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (4) : 409-460. doi: 10.3934/puqr.2021020

[5]

Xiaofan Guo, Shan Li, Xinpeng Li. On the laws of the iterated logarithm with mean-uncertainty under sublinear expectations. Probability, Uncertainty and Quantitative Risk, 2022, 7 (1) : 1-12. doi: 10.3934/puqr.2022001

[6]

Siyuan Tang. New time-changes of unipotent flows on quotients of Lorentz groups. Journal of Modern Dynamics, 2022, 18: 13-67. doi: 10.3934/jmd.2022002

[7]

Maria José Pacifico, Fan Yang. Hitting times distribution and extreme value laws for semi-flows. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5861-5881. doi: 10.3934/dcds.2017255

[8]

Chihurn Kim, Dong Han Kim. On the law of logarithm of the recurrence time. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 581-587. doi: 10.3934/dcds.2004.10.581

[9]

Armengol Gasull, Francesc Mañosas. Subseries and signed series. Communications on Pure and Applied Analysis, 2019, 18 (1) : 479-492. doi: 10.3934/cpaa.2019024

[10]

Santos González, Llorenç Huguet, Consuelo Martínez, Hugo Villafañe. Discrete logarithm like problems and linear recurring sequences. Advances in Mathematics of Communications, 2013, 7 (2) : 187-195. doi: 10.3934/amc.2013.7.187

[11]

Yongjiang Guo, Yuantao Song. The (functional) law of the iterated logarithm of the sojourn time for a multiclass queue. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1049-1076. doi: 10.3934/jimo.2018192

[12]

María Jesús Carro, Carlos Domingo-Salazar. The return times property for the tail on logarithm-type spaces. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 2065-2078. doi: 10.3934/dcds.2018084

[13]

Cheng Zheng. Sparse equidistribution of unipotent orbits in finite-volume quotients of $\text{PSL}(2,\mathbb R)$. Journal of Modern Dynamics, 2016, 10: 1-21. doi: 10.3934/jmd.2016.10.1

[14]

Constantine M. Dafermos. Hyperbolic balance laws with relaxation. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4271-4285. doi: 10.3934/dcds.2016.36.4271

[15]

Jon Chaika, Rodrigo Treviño. Logarithmic laws and unique ergodicity. Journal of Modern Dynamics, 2017, 11: 563-588. doi: 10.3934/jmd.2017022

[16]

Avner Friedman. Conservation laws in mathematical biology. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081

[17]

Mauro Garavello. A review of conservation laws on networks. Networks and Heterogeneous Media, 2010, 5 (3) : 565-581. doi: 10.3934/nhm.2010.5.565

[18]

Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010

[19]

Mauro Garavello, Roberto Natalini, Benedetto Piccoli, Andrea Terracina. Conservation laws with discontinuous flux. Networks and Heterogeneous Media, 2007, 2 (1) : 159-179. doi: 10.3934/nhm.2007.2.159

[20]

Ferenc A. Bartha, Hans Z. Munthe-Kaas. Computing of B-series by automatic differentiation. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 903-914. doi: 10.3934/dcds.2014.34.903

2020 Impact Factor: 0.848

Metrics

  • PDF downloads (49)
  • HTML views (132)
  • Cited by (2)

Other articles
by authors

[Back to Top]