Hypercyclic abelian semigroups of matrices on RnRn

In this paper, we bring together results about the existence of a somewhere dense (resp. dense) orbit and the minimal number of generators for abelian semigroups of matrices on RnRn. We solve the problem of determining the minimal number of matrices in normal form over RR which form a hypercyclic abelian semigroup on RnRn. In particular, we show that no abelian semigroup generated by [n+12] matrices on RnRn can be hypercyclic ([] denotes the integer part).