Dynamics of tuples of matrices in Jordan form

TL;DR

This paper establishes that at least n+1 matrices in Jordan form are needed to form a hypercyclic tuple over R^n, answering a specific open question in the dynamics of matrix tuples.

Contribution

It proves that the minimal size of a hypercyclic tuple of matrices in Jordan form over R is n+1, resolving an open problem in the field.

Findings

Minimum number of Jordan form matrices for hypercyclicity is n+1

Provides a definitive answer to an open question in matrix dynamics

Advances understanding of hypercyclic behavior in matrix tuples

Abstract

A tuple (T_1,...,T_k) of (n x n) matrices over R is called hypercyclic if for some x in R^n the set {T^{m_1} T^{m_2}...T^{m_k} x : m_1,m_2,...,m_k in N} is dense in R^n. We prove that the minimum number of (n x n) matrices in Jordan form over R which form a hypercyclic tuple is n+1. This answers a question of Costakis, Hadjiloucas and Manoussos.