Japanese Journal of Mathematics

, Volume 7, Issue 2, pp 135–166

Introduction to random walks on homogeneous spaces


    • Centre national de la recherche scientifique–Département de MathématiquesUniversité Paris-Sud 11
  • Jean-François Quint
    • Centre national de la recherche scientifique–Université Paris-Nord
Special Feature: The 10th Takagi Lectures

DOI: 10.1007/s11537-012-1220-9

Cite this article as:
Benoist, Y. & Quint, J. Jpn. J. Math. (2012) 7: 135. doi:10.1007/s11537-012-1220-9


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

Copyright information

© The Mathematical Society of Japan and Springer Japan 2012