Hypercyclic Abelian Semigroups of Matrices on ${\mathbb{R}}^n$

Abstract

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 ${\mathbb{R}}^n$. We solve the problem of determining the minimal number of matrices in normal form over $\mathbb{R}$ which form a hypercyclic abelian semigroup on R^n. In particular, we show that no abelian semigroup generated by $[\frac{n+1}{2}]$ matrices on ${\mathbb{R}}^n$ can be hypercyclic. ([ ] denotes the integer part). This is a corrected version of the paper published in Topology and its Applications 210 (2016), 29-45 (see also [4]). The differences between this version and the published version are explained at the end of the Introduction.

24 pages