Semigroup actions on tori and stationary measures on projective spaces

Volume 171 / 2005

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


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.


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

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image