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.
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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.
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Benoist, Y., Quint, JF. Introduction to random walks on homogeneous spaces. Jpn. J. Math. 7, 135–166 (2012). https://doi.org/10.1007/s11537-012-1220-9
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DOI: https://doi.org/10.1007/s11537-012-1220-9