Linear semigroups with coarsely dense orbits

TL;DR

This paper proves that coarsely dense orbits of finitely generated abelian semigroups of invertible linear operators are actually dense, providing a detailed analysis of orbit closures in real and complex vector spaces.

Contribution

It establishes that coarsely dense orbits are dense in the entire space and characterizes orbit closures in real vector spaces.

Findings

Coarsely dense orbits are dense in the vector space.

In the complex case, orbits are dense in the entire space.

Provides detailed descriptions of orbit closures in real vector spaces.

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.