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

Let G be an abelian semigroup of matrices on K n ( K = C or 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 ≥ 3 , but it can be minimal for n = 1 ; for K = R , the critical number is n = 2 . [ABSTRACT FROM AUTHOR]