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Applied Mathematics and Nonlinear Sciences

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Available online: December 08, 2017

J-class abelian semigroups of matrices on $\mathbb{R}^{n}$

Habib Marzougui
Volume 2 - Issue 2.     Year 2017.     Pages 519–528.

DOI: 10.21042/AMNS.2017.2.00043

Abstract

We establish, for finitely generated abelian semigroups $G$ of matrices on $\mathbb{R}^{n}$, and by using the extended limit sets $($the J-sets$)$, the following equivalence analogous to the complex case: $($i$)$ $G$ is hypercyclic, $($ii$)$ $J_{G}(v_{\eta}) = \mathbb{R}^{n}$ for some vector $v_{\eta}$ given by the structure of $G$, $($iii$)$ $\overline{G(v_{\eta})} = \mathbb{R}^{n}$. This answer a question raised by the author.Moreover we construct for any $n \geq 2$ an abelian semigroup $G$ of GL$(n, \mathbb{R})$ generated by $n+1$ diagonal matrices which is locally hypercyclic $($or J-class$)$ but not hypercyclic and such that J$_{G}(e_{k}) = \mathbb{R}^{n}$ for every $k= 1,\dots, n$, where $(e_{1},\dots, e_{n})$ is the canonical basis of $\mathbb{R}^{n}$. This gives a negative answer to a question raised by Costakis and Manoussos.

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ISSN:
2444-8656
Category:
Applied Mathematics, Nonlinear Sciences

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