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 a0 and a1 be two matrices in SL(2, \({\mathbb{Z}}\)) which span a non-solvable group. Let x0 be an irrational point on the torus \({\mathbb{T}^2}\). We toss a0 or a1, apply it to x0, get another irrational point x1, do it again to x1, get a point x2, 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 groupsdiscrete subgroupshomogeneous dynamicsrandom walk

Mathematics Subject Classification (2010)


Copyright information

© The Mathematical Society of Japan and Springer Japan 2012