Dynamics of abelian subgroups of GL(n, C): a structure's Theorem

Abstract

In this paper, we characterize the dynamic of every abelian subgroups $\mathcal{G}$ of GL($n$, $\mathbb{K}$), $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. We show that there exists a $\mathcal{G}$-invariant, dense open set $U$ in ${\mathbb{K}}^n$ saturated by minimal orbits with ${\mathbb{K}}^n-U$ a union of at most $n$ $\mathcal{G}$-invariant vectorial subspaces of ${\mathbb{K}}^n$ of dimension $n-1$ or $n-2$ on $\mathbb{K}$. As a consequence, $\mathcal{G}$ has height at most $n$ and in particular it admits a minimal set in ${\mathbb{K}}^n-\{0\}$.

16 pages