Special Feature: The 10th Takagi Lectures

Japanese Journal of Mathematics

, Volume 7, Issue 2, pp 135-166

First online:

Introduction to random walks on homogeneous spaces

  • Yves BenoistAffiliated withCentre national de la recherche scientifique–Département de Mathématiques, Université Paris-Sud 11 Email author 
  • , Jean-François QuintAffiliated withCentre national de la recherche scientifique–Université Paris-Nord

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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