Topological transitive Abelian subgrouns of GL(n,R)

TL;DR

This paper characterizes abelian subgroups of GL(n, R) with dense orbits in R^n, providing explicit criteria for finitely generated cases and showing limitations on the number of generators for dense orbits.

Contribution

It offers a complete characterization of abelian subgroups with dense orbits and establishes bounds on generators needed for such orbits.

Findings

Explicit characterization for finitely generated subgroups.

No abelian subgroup generated by less than or equal to (n+1)/2 matrices can have a dense orbit.

Provides conditions for locally dense and dense orbits in R^n.

Abstract

We give a complete characterization of abelian subgroups of GL(n, R) with a locally dense (resp. dense) orbit in R^n. For finitely generated subgroups, this characterization is explicit and it is used to show that no abelian subgroup of GL(n, R) generated by [ (n+1)/2 ] matrices can have a dense orbit in R^n. ([ ] denotes the integer part).