Semigroups of matrices with dense orbits

We prove that for any n ≥ 1 there exist n × n matrices A and B such that for any vector x ∈ R with a nonzero first component, the orbit of x under the action of the semigroup generated by A and B is dense in R. As a corollary, we prove that for a large set of diagonal matrices A and B and any vector V with nonzero entries, the orbit of any vector under the semigroup generated by the affine maps x → Ax + V and x → Bx is dense in R. 1 Main statements Let X be a topological vector space and T : X → X be a continuous linear operator on X. Then T is called hypercyclic if there exists a vector x ∈ X whose orbit {x, Tx, T x, . . .} is dense inX. It is a beautiful result by Bourdan and Feldman [3], in the locally convex case, and Wengenroth [7], in the general case, that if the orbit of x ∈ X is somewhere dense then for every nonzero polynomial p the operator p(T ) is hypercyclic. In [1], Ansari proved that all infinite-dimensional separable Banach spaces admit hypercyclic operators. On the other hand, in the finite dimensional case (i.e. X is isomorphic to a Euclidean space), no linear operator is hypercyclic. This can be seen by looking at the Jordan normal form of the matrix of the operator. It is then natural to ask whether one can find a pair of matrices A and B so that the orbit of some x ∈ X under the action of the semigroup generated by A and B is dense in X. In this paper, we prove the