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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. 

[2]

H. Abels and G. Margulis, preprint.

[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]

H. Abels and G. Margulis, preprint.

[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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