Minimality and non-existence of non-zero finite orbits for abelian linear semigroups

Let G be an abelian semigroup of matrices on \({\mathbb {K}}^{n}\) ${\mathbb{K}}^n$ ( \({\mathbb {K}}={\mathbb {C}}\) $\mathbb{K}=\mathbb{C}$ or \({\mathbb {R}}\) $\mathbb{R}$ ). We show that if G is hypercyclic, then it has no non-zero finite orbit. This result fails if we drop the assumption that G is abelian. As a consequence, if G is abelian, it is not chaotic. On the other hand, we show that G is not minimal for \(n\ge 3\) $n\ge 3$ , but it can be minimal for \(n=1\) $n=1$ ; for \({\mathbb {K}}={\mathbb {R}}\) $\mathbb{K}=\mathbb{R}$ , the critical number is \(n=2\) $n=2$ .