Linear semigroups with coarsely dense orbits

Abstract

Abstract. Let S be a finitely generated abelian semigroup of invertible linear operators on a finite dimensional real or complex vector space V . We show that every coarsely dense orbit of S is actually dense in V . More generally, if the orbit contains a coarsely dense subset of some open cone C in V then the closure of the orbit contains the closure of C. In the complex case the orbit is then actually dense in V . For the real case we give precise information about the possible cases for the closure of the orbit.

Topological generators of abelian Lie groups and hypercyclic finitely generated abelian semigroups of matrices

“…We will obtain more precise information about the closure of the orbit Sx in [1]. In particular we will prove that Gx is dense in V for the complex case, which implies the following.…”

“…In a forthcoming paper [1] we will give more precise information on the closure of Sx, if Sx is somewhere dense. It is a union of convex cones of a very special type.…”

“…In the same work Feldman gave a criterion for when a somewhere dense orbit is dense in terms of spectral properties of the elements of the semigroup [11,Theorem 5.5]. In section 5 we give an explanation for this difference between the real and the complex case based on a structure theorem for algebraic groups and in a forthcoming paper [1] will give more details. The contents of the paper are as follows.…”

Coarse topological transitivity on open cones and coarsely J -class and D -class operators

“…Coarsely dense orbits appear naturally by looking at perturbations of a dense orbit by a vector with bounded orbit [13]. In [1] we studied, together with H. Abels, a similar problem for finitely generated abelian subsemigroups of GL(V ), where V is a finite dimensional complex (or real) vector space. We showed that if such a semigroup has a coarsely dense orbit on an open cone of V then, for the complex case, this orbit is actually dense in V .…”