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Logarithm laws for unipotent flows, Ⅱ
1. | Department of Mathematics, University of Washington, Seattle, WA 98195, USA |
2. | Department of Mathematics, Yale University, New Haven, CT 06520, USA |
We prove analogs of the logarithm laws of Sullivan and KleinbockMargulis in the context of unipotent flows. In particular, we prove results for horospherical actions on homogeneous spaces G/Γ.
References:
[1] |
H. Abels and G. Margulis,
Coarsely geodesic metrics on reductive groups, in Modern Dynamical Systems and Applications, Cambridge Univ. Press, Cambridge, (2004), 163-183.
|
[2] | |
[3] |
J. S. Athreya,
Cusp excursions on parameter spaces, J. London Math. Soc., 87 (2013), 741-765.
doi: 10.1112/jlms.2013.87.issue-3. |
[4] |
J. S. Athreya and Y. Cheung,
A Poincaré section for horocycle flow on the space of lattices, Int. Math. Res. Notices, no. 10 (2014), 2643-2690.
doi: 10.1093/imrn/rnt003. |
[5] |
J. S. Athreya and G. Margulis,
Logarithm laws for unipotent flows, I, Journal of Modern Dynamics, 3 (2009), 359-378.
doi: 10.3934/jmd.2009.3.359. |
[6] |
J. S. Athreya and F. Paulin,
Logarithm laws for strong unstable foliations in negative curvature and non-Archimedean Diophantine approximation, Groups, Geometry, and Dynamics, 8 (2014), 285-309.
doi: 10.4171/GGD/226. |
[7] |
A. Borel, Linear Algebraic Groups, 2nd enlarged edition, Springer Verlag, New York, 1991.
doi: 10.1007/978-1-4612-0941-6. |
[8] |
S. G. Dani,
Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math., 359 (1985), 55-89.
doi: 10.1515/crll.1985.359.55. |
[9] |
W. Feller,
An Introduction to Probability Theory and Its Applications, 1, Wiley, (1957).
|
[10] |
H. Garland and M. S. Raghunathan,
Fundamental domains for lattices in (R-)rank 1 semisimple Lie groups, Annals of Math. (2), 92 (1970), 279-326.
doi: 10.2307/1970838. |
[11] |
J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972. |
[12] |
J. Humphreys, Linear Algebraic Groups, 2nd printing, Springer-Verlag, New York-Heidelberg, 1975. |
[13] |
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. |
[14] |
D. Y. Kleinbock and G. A. Margulis,
Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494.
doi: 10.1007/s002220050350. |
[15] |
E. Leuzinger,
Geodesic rays in locally symmetric spaces, Differential Geometry and its Applications, 6 (1996), 55-65.
doi: 10.1016/0926-2245(96)00007-1. |
[16] |
G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, Berlin-New York, 1991.
doi: 10.1007/978-3-642-51445-6. |
[17] |
C. C. Moore,
Ergodicity of flows on homogeneous spaces, Amer. J. Math., 88 (1966), 154-178.
doi: 10.2307/2373052. |
[18] |
G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Annals of Math. Studies, Princeton Univ. Press, 1973. |
[19] |
D. Sullivan,
Disjoint spheres, approximation by quadratic numbers and the logarithm law for geodesics, Acta Mathematica, 149 (1982), 215-237.
doi: 10.1007/BF02392354. |
[20] |
B. Weiss,
Divergent trajectories on noncompact parameter spaces, Geom. and Funct. Anal., 14 (2004), 94-149.
doi: 10.1007/s00039-004-0453-z. |
show all references
References:
[1] |
H. Abels and G. Margulis,
Coarsely geodesic metrics on reductive groups, in Modern Dynamical Systems and Applications, Cambridge Univ. Press, Cambridge, (2004), 163-183.
|
[2] | |
[3] |
J. S. Athreya,
Cusp excursions on parameter spaces, J. London Math. Soc., 87 (2013), 741-765.
doi: 10.1112/jlms.2013.87.issue-3. |
[4] |
J. S. Athreya and Y. Cheung,
A Poincaré section for horocycle flow on the space of lattices, Int. Math. Res. Notices, no. 10 (2014), 2643-2690.
doi: 10.1093/imrn/rnt003. |
[5] |
J. S. Athreya and G. Margulis,
Logarithm laws for unipotent flows, I, Journal of Modern Dynamics, 3 (2009), 359-378.
doi: 10.3934/jmd.2009.3.359. |
[6] |
J. S. Athreya and F. Paulin,
Logarithm laws for strong unstable foliations in negative curvature and non-Archimedean Diophantine approximation, Groups, Geometry, and Dynamics, 8 (2014), 285-309.
doi: 10.4171/GGD/226. |
[7] |
A. Borel, Linear Algebraic Groups, 2nd enlarged edition, Springer Verlag, New York, 1991.
doi: 10.1007/978-1-4612-0941-6. |
[8] |
S. G. Dani,
Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math., 359 (1985), 55-89.
doi: 10.1515/crll.1985.359.55. |
[9] |
W. Feller,
An Introduction to Probability Theory and Its Applications, 1, Wiley, (1957).
|
[10] |
H. Garland and M. S. Raghunathan,
Fundamental domains for lattices in (R-)rank 1 semisimple Lie groups, Annals of Math. (2), 92 (1970), 279-326.
doi: 10.2307/1970838. |
[11] |
J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972. |
[12] |
J. Humphreys, Linear Algebraic Groups, 2nd printing, Springer-Verlag, New York-Heidelberg, 1975. |
[13] |
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. |
[14] |
D. Y. Kleinbock and G. A. Margulis,
Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494.
doi: 10.1007/s002220050350. |
[15] |
E. Leuzinger,
Geodesic rays in locally symmetric spaces, Differential Geometry and its Applications, 6 (1996), 55-65.
doi: 10.1016/0926-2245(96)00007-1. |
[16] |
G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, Berlin-New York, 1991.
doi: 10.1007/978-3-642-51445-6. |
[17] |
C. C. Moore,
Ergodicity of flows on homogeneous spaces, Amer. J. Math., 88 (1966), 154-178.
doi: 10.2307/2373052. |
[18] |
G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Annals of Math. Studies, Princeton Univ. Press, 1973. |
[19] |
D. Sullivan,
Disjoint spheres, approximation by quadratic numbers and the logarithm law for geodesics, Acta Mathematica, 149 (1982), 215-237.
doi: 10.1007/BF02392354. |
[20] |
B. Weiss,
Divergent trajectories on noncompact parameter spaces, Geom. and Funct. Anal., 14 (2004), 94-149.
doi: 10.1007/s00039-004-0453-z. |
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