Semigroup actions on tori and stationary measures on projective spaces
Copyright policy )Semigroup actions on tori and stationary measures on projective spaces
Let $��$ be a sub-semigroup of $G=GL(d,\mathbb R),$ $d>1.$ We assume that the action of $��$ on $\R^d$ is strongly irreducible and that $��$ contains a proximal and expanding element. We describe contraction properties of the dynamics of $��$ on $\R^d$ at infinity. This amounts to the consideration of the action of $��$ on some compact homogeneous spaces of $G,$ which are extensions of the projective space $\pr^{d-1}.$ In the case where $��$ is a sub-semigroup of $GL(d,\R)\cap M(d,\Z)$ and $��$ has the above properties, we deduce that the $��$-orbits on $\T^d=\R^d\slash\Z^d$ are finite or dense.
- University of Wrocław Poland
- Université de Rennes 1 France
- Mathematics Research Institute of Rennes France
- University of Rennes 2 France
- University of Rennes France
limit set, 54H20 ; 22E40 ; 60J05 ; 60B15, asymptotic set, Probability theory on linear topological spaces, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], [MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS], Dynamical Systems (math.DS), Group Theory (math.GR), Topological dynamics, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], 510, random walk, 60J05, FOS: Mathematics, proximal and expanding element, ID-property, Mathematics - Dynamical Systems, 22E40, [MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR], General groups of measure-preserving transformations and dynamical systems, Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\), projective space, proximal and quasi-expanding element, stationary measure, General theory of group and pseudogroup actions, 54H20, toral automorphism, Mathematics - Group Theory, 60B15
limit set, 54H20 ; 22E40 ; 60J05 ; 60B15, asymptotic set, Probability theory on linear topological spaces, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], [MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS], Dynamical Systems (math.DS), Group Theory (math.GR), Topological dynamics, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], 510, random walk, 60J05, FOS: Mathematics, proximal and expanding element, ID-property, Mathematics - Dynamical Systems, 22E40, [MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR], General groups of measure-preserving transformations and dynamical systems, Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\), projective space, proximal and quasi-expanding element, stationary measure, General theory of group and pseudogroup actions, 54H20, toral automorphism, Mathematics - Group Theory, 60B15
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