On multi-hypercyclic abelian semigroups of matrices on $\mathbb{R}^{n}$

Abstract

We give a complete characterization of existence of dense orbit for any abelian semigroup of matrices on R n . For finitely generated semigroups, this characterization is explicit and it is used to determine the minimal number of matrices in normal form over R which form a hypercyclic abelian semigroup on R n . In particular, we show that no abelian semigroup generated by n+1 2 matrices on R n can be hypercyclic. ([ ] denotes the integer part).2000 Mathematics Subject Classification. 37C85, 47A16.