On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple

TL;DR

This paper introduces the concept of locally hypercyclic tuples of operators, showing that such tuples can exist with fewer matrices than hypercyclic ones, and establishes the minimal number of diagonal matrices needed for hypercyclicity.

Contribution

It extends hypercyclicity to locally hypercyclic tuples, demonstrating their existence with fewer matrices and determining the minimal number of diagonal matrices for hypercyclicity.

Findings

Existence of locally hypercyclic, non-hypercyclic tuples in finite-dimensional spaces.

Minimal number of diagonal matrices for hypercyclicity in $ extbf{R}^n$ is $n+1$.

Contrast between locally hypercyclic and hypercyclic tuples regarding matrix count.

Abstract

In this paper we extend the notion of a locally hypercyclic operator to that of a locally hypercyclic tuple of operators. We then show that the class of hypercyclic tuples of operators forms a proper subclass to that of locally hypercyclic tuples of operators. What is rather remarkable is that in every finite dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$, a pair of commuting matrices exists which forms a locally hypercyclic, non-hypercyclic tuple. This comes in direct contrast to the case of hypercyclic tuples where the minimal number of matrices required for hypercyclicity is related to the dimension of the vector space. In this direction we prove that the minimal number of diagonal matrices required to form a hypercyclic tuple on $\mathbb{R}^n$ is $n+1$, thus complementing a recent result due to Feldman.