J-Class Abelian Semigroups of Matrices on C^n and Hypercyclicity

TL;DR

This paper characterizes hypercyclic abelian semigroups of matrices on complex n-space using J-sets and constructs examples that are locally hypercyclic but not hypercyclic, answering a previously open question.

Contribution

It provides a new characterization of hypercyclicity for abelian matrix semigroups and constructs explicit examples illustrating the distinction between local hypercyclicity and hypercyclicity.

Findings

Characterization of hypercyclic abelian semigroups using extended limit sets.

Construction of semigroups that are locally hypercyclic but not hypercyclic.

Negative answer to a question by Costakis and Manoussos.

Abstract

We give a characterization of hypercyclic finitely generated abelian semigroups of matrices on C^n using the extended limit sets (the J-sets). Moreover we construct for any n\geq 2 an abelian semigroup G of GL(n;C) generated by n + 1 diagonal matrices which is locally hypercyclic but not hypercyclic and such that JG(e_k) = C^n for every k = 1; : : : ; n, where (e_1; : : : ; e_n) is the canonical basis of C^n. This gives a negative answer to a question raised by Costakis and Manoussos.