Dynamics of tuples of matrices

TL;DR

This paper investigates the dynamics of tuples of matrices, proving the existence of hypercyclic tuples of matrices over real numbers for any dimension, and explores related Jordan form cases.

Contribution

It demonstrates the existence of hypercyclic tuples of matrices for all dimensions and provides results for Jordan form matrices over real and complex fields.

Findings

Existence of hypercyclic tuples for all n ≥ 2

Construction of hypercyclic tuples in Jordan form

Results applicable to both real and complex matrices

Abstract

In this article we answer a question raised by N. Feldman in \cite{Feldman} concerning the dynamics of tuples of operators on $\mathbb{R}^n$. In particular, we prove that for every positive integer $n\geq 2$ there exist $n$ tuples $(A_1, A_2, ..., A_n)$ of $n\times n$ matrices over $\mathbb{R}$ such that $(A_1, A_2, ..., A_n)$ is hypercyclic. We also establish related results for tuples of $2\times 2$ matrices over $\mathbb{R}$ or $\mathbb{C}$ being in Jordan form.