Semigroup actions on tori and stationary measures on projective spaces

Yves Guivarc'h; Roman Urban

Studia Mathematica (2005)

  • Volume: 171, Issue: 1, page 33-66
  • ISSN: 0039-3223

Abstract

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

How to cite

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Yves Guivarc'h, and Roman Urban. "Semigroup actions on tori and stationary measures on projective spaces." Studia Mathematica 171.1 (2005): 33-66. <http://eudml.org/doc/286295>.

@article{YvesGuivarch2005,
abstract = {Let Γ be a subsemigroup of G = GL(d,ℝ), d > 1. We assume that the action of Γ on $ℝ^\{d\}$ is strongly irreducible and that Γ contains a proximal and quasi-expanding element. We describe contraction properties of the dynamics of Γ on $ℝ^\{d\}$ at infinity. This amounts to the consideration of the action of Γ on some compact homogeneous spaces of G, which are extensions of the projective space $ℙ^\{d-1\}$. In the case where Γ is a subsemigroup of GL(d,ℝ) ∩ M(d,ℤ) and Γ has the above properties, we deduce that the Γ-orbits on $^\{d\} = ℝ^\{d\}/ℤ^\{d\}$ are finite or dense.},
author = {Yves Guivarc'h, Roman Urban},
journal = {Studia Mathematica},
keywords = {asymptotic set; proximal and quasi-expanding element; toral automorphism; ID-property; random walk; projective space; stationary measure},
language = {eng},
number = {1},
pages = {33-66},
title = {Semigroup actions on tori and stationary measures on projective spaces},
url = {http://eudml.org/doc/286295},
volume = {171},
year = {2005},
}

TY - JOUR
AU - Yves Guivarc'h
AU - Roman Urban
TI - Semigroup actions on tori and stationary measures on projective spaces
JO - Studia Mathematica
PY - 2005
VL - 171
IS - 1
SP - 33
EP - 66
AB - Let Γ be a subsemigroup of G = GL(d,ℝ), d > 1. We assume that the action of Γ on $ℝ^{d}$ is strongly irreducible and that Γ contains a proximal and quasi-expanding element. We describe contraction properties of the dynamics of Γ on $ℝ^{d}$ at infinity. This amounts to the consideration of the action of Γ on some compact homogeneous spaces of G, which are extensions of the projective space $ℙ^{d-1}$. In the case where Γ is a subsemigroup of GL(d,ℝ) ∩ M(d,ℤ) and Γ has the above properties, we deduce that the Γ-orbits on $^{d} = ℝ^{d}/ℤ^{d}$ are finite or dense.
LA - eng
KW - asymptotic set; proximal and quasi-expanding element; toral automorphism; ID-property; random walk; projective space; stationary measure
UR - http://eudml.org/doc/286295
ER -

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