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Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms

Published online by Cambridge University Press:  04 May 2004

Y. GUIVARC'H  and
A. N. STARKOV
Y. GUIVARC'H
Affiliation:
Institut de Recherche Mathématiques de Rennes 1, Campus de Beaulieu 35042, Rennes Cedex, France (e-mail: [email protected])
A. N. STARKOV
Affiliation:
All-Russian Institute of Electrotechnics, 143500, Istra, Moscow Region, Russia and Department of Mechanics and Mathematics, Moscow State University, 117234, Moscow, Russia (e-mail: [email protected])
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Abstract

Let V be a finite-dimensional vector space over $\mathbb{R}$and let $\Gamma \subset \mathrm{GL}(V)$ be a semigroup. We study the closed $\Gamma$-invariant subsets of V-\{0\} under the condition that the Zariski closure of $\Gamma$ is semi-simple. We use the results to show that, if $\Gamma\subset\mathrm{SL}(\mathbb{R}^d)$ acts on $\mathbb{T}^d$ by automorphisms, then the orbits of $\Gamma$ are finite or dense.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 24 , Issue 3 , June 2004 , pp. 767 - 802
Copyright
2004 Cambridge University Press

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