环面上的半群作用与射影空间上的平稳测度
Semigroup actions on tori and stationary measures on projective spaces
摘要：
设$Γ$是$G=GL(d,\mathbb R),$ $d>1.$的一个子半群.我们假设$Γ$在$\R^d$上的作用是强不可约的，并且$Γ$包含一个邻近扩张元.我们描述了在无穷远处的$\R^d$上的$Γ$动力学的收缩性质。这相当于考虑$Γ$在$G,$的某些紧致齐性空间上的作用，这些空间是射影空间$\pr^{d-1}.$的扩张.当$Γ$是$GL(d,\R)\cap M(d,\Z)$的子半群且$Γ$具有上述性质时，我们推出了$\T^d=\R^d\slash\Z^d$上的$Γ$ -轨道是有限的或稠密的.
英文摘要：
Let $Γ$ be a sub-semigroup of $G=GL(d,\mathbb R),$ $d>1.$ We assume that the action of $Γ$ on $\R^d$ is strongly irreducible and that $Γ$ contains a proximal and 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 $\pr^{d-1}.$ In the case where $Γ$ is a sub-semigroup of $GL(d,\R)\cap M(d,\Z)$ and $Γ$ has the above properties, we deduce that the $Γ$-orbits on $\T^d=\R^d\slash\Z^d$ are finite or dense.
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