American Institute of Mathematical Sciences

Advanced
  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

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. 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

[1]

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
[2]

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

Shucheng Yu. Logarithm laws for unipotent flows on hyperbolic manifolds. Journal of Modern Dynamics, 2017, 11: 447-476. doi: 10.3934/jmd.2017018
[4]

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
[5]

Yongjiang Guo, Yuantao Song. The (functional) law of the iterated logarithm of the sojourn time for a multiclass queue. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-28. doi: 10.3934/jimo.2018192
[6]

Alberto Bressan, Graziano Guerra. Shift-differentiabilitiy of the flow generated by a conservation law. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 35-58. doi: 10.3934/dcds.1997.3.35
[7]

Alberto Bressan, Khai T. Nguyen. Conservation law models for traffic flow on a network of roads. Networks & Heterogeneous Media, 2015, 10 (2) : 255-293. doi: 10.3934/nhm.2015.10.255
[8]

Lukas F. Lang, Otmar Scherzer. Optical flow on evolving sphere-like surfaces. Inverse Problems & Imaging, 2017, 11 (2) : 305-338. doi: 10.3934/ipi.2017015
[9]

Michael Herty, Lorenzo Pareschi, Mohammed Seaïd. Enskog-like discrete velocity models for vehicular traffic flow. Networks & Heterogeneous Media, 2007, 2 (3) : 481-496. doi: 10.3934/nhm.2007.2.481
[10]

Anna Marciniak-Czochra, Andro Mikelić. A nonlinear effective slip interface law for transport phenomena between a fracture flow and a porous medium. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1065-1077. doi: 10.3934/dcdss.2014.7.1065
[11]

J. S. Athreya, Anish Ghosh, Amritanshu Prasad. Ultrametric logarithm laws I. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 337-348. doi: 10.3934/dcdss.2009.2.337
[12]

Asim Aziz, Wasim Jamshed. Unsteady MHD slip flow of non Newtonian power-law nanofluid over a moving surface with temperature dependent thermal conductivity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 617-630. doi: 10.3934/dcdss.2018036
[13]

Gildas Besançon, Didier Georges, Zohra Benayache. Towards nonlinear delay-based control for convection-like distributed systems: The example of water flow control in open channel systems. Networks & Heterogeneous Media, 2009, 4 (2) : 211-221. doi: 10.3934/nhm.2009.4.211
[14]

Afaf Bouharguane. On the instability of a nonlocal conservation law. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 419-426. doi: 10.3934/dcdss.2012.5.419
[15]

JÓzsef Balogh, Hoi Nguyen. A general law of large permanent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5285-5297. doi: 10.3934/dcds.2017229
[16]

María Jesús Carro, Carlos Domingo-Salazar. The return times property for the tail on logarithm-type spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2065-2078. doi: 10.3934/dcds.2018084
[17]

Michał Misiurewicz, Sonja Štimac. Lozi-like maps. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2965-2985. doi: 10.3934/dcds.2018127
[18]

Ambroise Vest. On the structural properties of an efficient feedback law. Evolution Equations & Control Theory, 2013, 2 (3) : 543-556. doi: 10.3934/eect.2013.2.543
[19]

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
[20]

Michal Kupsa, Štěpán Starosta. On the partitions with Sturmian-like refinements. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3483-3501. doi: 10.3934/dcds.2015.35.3483

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (28)
  • HTML views (78)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences