, Volume 7, Issue 2, pp 135-166
Date: 17 Nov 2012

Introduction to random walks on homogeneous spaces

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Abstract

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.

Communicated by: Toshiyuki Kobayashi
This article is based on the 10th Takagi Lectures that the author delivered at Research Institute for Mathematical Sciences, Kyoto University on May 26, 2012.