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

TL;DR

This paper investigates the minimal number of generators needed for finitely generated abelian semigroups of matrices to have dense orbits, linking algebraic group actions with topological density properties.

Contribution

It provides a method to determine the minimal number of generators for dense orbits in abelian matrix semigroups by analyzing their Zariski closure.

Findings

Determined minimal generators for dense orbits in abelian matrix semigroups.

Connected component of Zariski closure influences generator count.

Unified approach linking algebraic and topological properties.

Abstract

In this paper we bring together results about the density of subsemigroups of abelian Lie groups, the minimal number of topological generators of abelian Lie groups and a result about actions of algebraic groups. We find the minimal number of generators of a finitely generated abelian semigroup or group of matrices with a dense or a somewhere dense orbit by computing the minimal number of generators of a dense subsemigroup (or subgroup) of the connected component of the identity of its Zariski closure.