Logarithm laws for unipotent flows, Ⅱ

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2017, 11: 1-16. doi: 10.3934/jmd.2017001

Logarithm laws for unipotent flows, Ⅱ

Jayadev S. Athreya ^1, and Gregory A. Margulis ^2,

1.	Department of Mathematics, University of Washington, Seattle, WA 98195, USA
2.	Department of Mathematics, Yale University, New Haven, CT 06520, USA

Received October 22, 2014 Revised August 30, 2016 Published December 2016

Fund Project: Supported by NSF grants DMS 0603636, DMS 1069153, and CAREER grant DMS 1351853.Supported by NSF grant DMS 0801195 and 1265695

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.

Keywords: Logarithm law, unipotent flow, norm-like pseudometric.

Mathematics Subject Classification: 37A17; Secondary: 11H16.

Citation: 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

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. Google Scholar
[2]	H. Abels and G. Margulis, preprint.Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	A. Borel, Linear Algebraic Groups, 2nd enlarged edition, Springer Verlag, New York, 1991. doi: 10.1007/978-1-4612-0941-6. Google Scholar
[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. Google Scholar
[9]	W. Feller, An Introduction to Probability Theory and Its Applications, 1, Wiley, (1957). Google Scholar
[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. Google Scholar
[11]	J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972. Google Scholar
[12]	J. Humphreys, Linear Algebraic Groups, 2nd printing, Springer-Verlag, New York-Heidelberg, 1975. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, Berlin-New York, 1991. doi: 10.1007/978-3-642-51445-6. Google Scholar
[17]	C. C. Moore, Ergodicity of flows on homogeneous spaces, Amer. J. Math., 88 (1966), 154-178. doi: 10.2307/2373052. Google Scholar
[18]	G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Annals of Math. Studies, Princeton Univ. Press, 1973. Google Scholar
[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. Google Scholar
[20]	B. Weiss, Divergent trajectories on noncompact parameter spaces, Geom. and Funct. Anal., 14 (2004), 94-149. doi: 10.1007/s00039-004-0453-z. Google Scholar

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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. Google Scholar
[2]	H. Abels and G. Margulis, preprint.Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	A. Borel, Linear Algebraic Groups, 2nd enlarged edition, Springer Verlag, New York, 1991. doi: 10.1007/978-1-4612-0941-6. Google Scholar
[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. Google Scholar
[9]	W. Feller, An Introduction to Probability Theory and Its Applications, 1, Wiley, (1957). Google Scholar
[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. Google Scholar
[11]	J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972. Google Scholar
[12]	J. Humphreys, Linear Algebraic Groups, 2nd printing, Springer-Verlag, New York-Heidelberg, 1975. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, Berlin-New York, 1991. doi: 10.1007/978-3-642-51445-6. Google Scholar
[17]	C. C. Moore, Ergodicity of flows on homogeneous spaces, Amer. J. Math., 88 (1966), 154-178. doi: 10.2307/2373052. Google Scholar
[18]	G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Annals of Math. Studies, Princeton Univ. Press, 1973. Google Scholar
[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. Google Scholar
[20]	B. Weiss, Divergent trajectories on noncompact parameter spaces, Geom. and Funct. Anal., 14 (2004), 94-149. doi: 10.1007/s00039-004-0453-z. Google Scholar

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