J-class abelian semigroups of matrices on $\mathbb{C }^{n}$ and hypercyclicity

Abstract

We give a characterization of hypercyclic finitely generated abelian semigroups of matrices on $\mathbb{C }^{n}$ using the extended limit sets (the J-sets). Moreover we construct for any $n\ge 2$ an abelian semigroup $G$ of GL$(n, \mathbb{C })$ generated by $n+1$ diagonal matrices which is locally hypercyclic but not hypercyclic and such that J$_{G}(e_{k}) = \mathbb{C }^{n}$ for every $k= 1,\dots , n$, where $(e_{1},\dots , e_{n})$ is the canonical basis of $\mathbb{C }^{n}$. This gives a negative answer to a question raised by Costakis and Manoussos.