American Institute of Mathematical Sciences

Advanced
July  2009, 3(3): 359-378. doi: 10.3934/jmd.2009.3.359

Logarithm laws for unipotent flows, I

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

1. 

Department of Mathematics, Yale University, New Haven, CT 06520-8283, United States, United States

Received  November 2008 Revised  May 2009 Published  August 2009

We prove analogs of the logarithm laws of Sullivan and Kleinbock--Margulis in the context of unipotent flows. In particular, we obtain results for one-parameter actions on the space of lattices SL(n, $\R$)/SL(n, $\Z$). The key lemma for our results says the measure of the set of unimodular lattices in $\R^n$ that does not intersect a 'large' volume subset of $\R^n$ is 'small'. This can be considered as a 'random' analog of the classical Minkowski Theorem in the geometry of numbers.
Keywords: Logarithm laws, geometry of numbers., unipotent flows, diophantine approximation.
Mathematics Subject Classification: Primary: 327A17; Secondary: 11H1.
Citation: 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
[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]

Shucheng Yu. Logarithm laws for unipotent flows on hyperbolic manifolds. Journal of Modern Dynamics, 2017, 11: 447-476. doi: 10.3934/jmd.2017018
[3]

Dmitry Kleinbock, Barak Weiss. Dirichlet's theorem on diophantine approximation and homogeneous flows. Journal of Modern Dynamics, 2008, 2 (1) : 43-62. doi: 10.3934/jmd.2008.2.43
[4]

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
[5]

Michael Björklund, Alexander Gorodnik. Central limit theorems in the geometry of numbers. Electronic Research Announcements, 2017, 24: 110-122. doi: 10.3934/era.2017.24.012
[6]

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
[7]

Shrikrishna G. Dani. Simultaneous diophantine approximation with quadratic and linear forms. Journal of Modern Dynamics, 2008, 2 (1) : 129-138. doi: 10.3934/jmd.2008.2.129
[8]

Chao Ma, Baowei Wang, Jun Wu. Diophantine approximation of the orbits in topological dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2455-2471. doi: 10.3934/dcds.2019104
[9]

Hans Koch, João Lopes Dias. Renormalization of diophantine skew flows, with applications to the reducibility problem. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 477-500. doi: 10.3934/dcds.2008.21.477
[10]

Zengjing Chen, Qingyang Liu, Gaofeng Zong. Weak laws of large numbers for sublinear expectation. Mathematical Control and Related Fields, 2018, 8 (3&4) : 637-651. doi: 10.3934/mcrf.2018027
[11]

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
[12]

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
[13]

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
[14]

Sanghoon Kwon, Seonhee Lim. Equidistribution with an error rate and Diophantine approximation over a local field of positive characteristic. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 169-186. doi: 10.3934/dcds.2018008
[15]

Yuliya Gorb, Dukjin Nam, Alexei Novikov. Numerical simulations of diffusion in cellular flows at high Péclet numbers. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 75-92. doi: 10.3934/dcdsb.2011.15.75
[16]

Boris Andreianov, Kenneth H. Karlsen, Nils H. Risebro. On vanishing viscosity approximation of conservation laws with discontinuous flux. Networks and Heterogeneous Media, 2010, 5 (3) : 617-633. doi: 10.3934/nhm.2010.5.617
[17]

Piotr Gwiazda, Piotr Orlinski, Agnieszka Ulikowska. Finite range method of approximation for balance laws in measure spaces. Kinetic and Related Models, 2017, 10 (3) : 669-688. doi: 10.3934/krm.2017027
[18]

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
[19]

Qing Yi. On the Stokes approximation equations for two-dimensional compressible flows. Kinetic and Related Models, 2013, 6 (1) : 205-218. doi: 10.3934/krm.2013.6.205
[20]

Anupam Sen, T. Raja Sekhar. Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation. Communications on Pure and Applied Analysis, 2019, 18 (2) : 931-942. doi: 10.3934/cpaa.2019045

2021 Impact Factor: 0.641

Tools

Metrics

  • PDF downloads (143)
  • HTML views (0)
  • Cited by (20)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy