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# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Semigroup actions on tori and stationary measures on projective spaces

### Tom 171 / 2005

Studia Mathematica 171 (2005), 33-66 MSC: 54H20, 37C85, 60B11, 60J05. DOI: 10.4064/sm171-1-3

#### Streszczenie

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.

#### Autorzy

• Yves Guivarc'hIRMAR
Université de Rennes 1
Campus de Beaulieu
35042 Rennes Cedex, France
e-mail
• Roman UrbanInstitute of Mathematics
Wroc/law University
Plac Grunwaldzki 2/4
50-384 Wroc/law, Poland
e-mail

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