Semigroups of matrices with dense orbits

TL;DR

This paper constructs examples of matrix semigroups over real or complex fields where, for most vectors, the generated orbit is dense in the entire space, illustrating dense orbit phenomena in linear dynamics.

Contribution

It provides explicit examples of matrix semigroups with dense orbits for almost all vectors, advancing understanding of linear dynamical systems and orbit density.

Findings

Existence of matrices with dense orbits for almost all vectors

Dense orbit behavior in matrix semigroups over real and complex fields

Explicit constructions demonstrating dense orbits

Abstract

We give examples of $n \times n$ matrices $A$ and $B$ over the filed $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ such that for almost every column vector $x \in \mathbb{K}^n$, the orbit of $x$ under the action of the semigroup generated by $A$ and $B$ is dense in $\mathbb{K}^n$.