Japanese Journal of Mathematics

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

Introduction to random walks on homogeneous spaces

Special Feature: The 10th Takagi Lectures

Abstract

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 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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