Semigroup actions on tori and stationary measures on projective spaces

Yves Guivarc'h¹ ; Roman Urban²

¹ IRMAR Université de Rennes 1 Campus de Beaulieu 35042 Rennes Cedex, France
² Institute of Mathematics Wroc/law University Plac Grunwaldzki 2/4 50-384 Wroc/law, Poland

Studia Mathematica, Tome 171 (2005) no. 1, pp. 33-66

Résumé

Let ${\mit\Gamma}$ be a subsemigroup of $G=\mathrm{GL}(d,\mathbb R),$ $d>1.$ We assume that the action of ${\mit\Gamma}$ on ${\mathbb R}^d$ is strongly irreducible and that ${\mit\Gamma}$ contains a proximal and quasi-expanding element. We describe contraction properties of the dynamics of ${\mit\Gamma}$ on ${\mathbb R}^d$ at infinity. This amounts to the consideration of the action of ${\mit\Gamma}$ on some compact homogeneous spaces of $G,$ which are extensions of the projective space ${\mathbb P}^{d-1}.$ In the case where ${\mit\Gamma}$ is a subsemigroup of $\mathrm{GL}(d,{\mathbb R})\cap\mathrm{M}(d,{\mathbb Z})$ and ${\mit\Gamma}$ has the above properties, we deduce that the ${\mit\Gamma}$-orbits on ${\mathbb T}^d={\mathbb R}^d/{\mathbb Z}^d$ are finite or dense.

Détail

DOI : 10.4064/sm171-1-3

Keywords: mit gamma subsemigroup mathrm mathbb assume action mit gamma mathbb strongly irreducible mit gamma contains proximal quasi expanding element describe contraction properties dynamics mit gamma mathbb infinity amounts consideration action mit gamma compact homogeneous spaces nbsp which extensions projective space mathbb d where mit gamma subsemigroup mathrm mathbb cap mathrm mathbb mit gamma has above properties deduce mit gamma orbits mathbb mathbb mathbb finite dense