Japanese Journal of Mathematics

, Volume 7, Issue 2, pp 135–166 | Cite as

Introduction to random walks on homogeneous spaces

  • Yves Benoist
  • Jean-François Quint
Special Feature: The 10th Takagi Lectures


Let a 0 and a 1 be two matrices in SL(2, \({\mathbb{Z}}\)) which span a non-solvable group. Let x 0 be an irrational point on the torus \({\mathbb{T}^2}\). We toss a 0 or a 1, apply it to x 0, get another irrational point x 1, do it again to x 1, get a point x 2, and again. This random trajectory is equidistributed on the torus. This phenomenon is quite general on any finite volume homogeneous space.

Keywords and phrases

Lie groups discrete subgroups homogeneous dynamics random walk 

Mathematics Subject Classification (2010)

22E40 37C85 60J05 


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Copyright information

© The Mathematical Society of Japan and Springer Japan 2012

Authors and Affiliations

  1. 1.Centre national de la recherche scientifique–Département de MathématiquesUniversité Paris-Sud 11Orsay CedexFrance
  2. 2.Centre national de la recherche scientifique–Université Paris-NordVilletaneuseFrance

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