<abstract xmlns="http://www.w3.org/1999/xhtml">We establish, for finitely generated abelian semigroups <italic>G</italic> of matrices on ℝ<italic>n</italic>, and by using the extended limit sets (the J-sets), the following equivalence analogous to the complex case: (i) <italic>G</italic> is hypercyclic, (ii) <italic>JG</italic>(<italic>v</italic>η) = ℝ<italic>n</italic> for some vector <italic>v</italic><italic>η</italic> given by the structure of <italic>G</italic>, (iii) <overline><italic>G</italic>(<italic>vη</italic>)</overline> = ℝ<italic>n</italic>. This answer a question raised by the author. Moreover we construct for any <italic>n</italic> = 2 an abelian semigroup <italic>G</italic> of GL(<italic>n</italic>, ℝ) generated by <italic>n</italic> + 1 diagonal matrices which is locally hypercyclic (or J-class) but not hypercyclic and such that J<italic>G</italic>(<italic>ek</italic>) = ℝ<italic>n</italic> for every <italic>k</italic> = 1,…, <italic>n</italic>, where (<italic>e</italic>1,…, <italic>e</italic><italic>n</italic>) is the canonical basis of ℝ<italic>n</italic>. This gives a negative answer to a question raised by Costakis and Manoussos</abstract>

We establish, for finitely generated abelian semigroups G of matrices on ℝn, and by using the extended limit sets (the J-sets), the following equivalence analogous to the complex case: (i) G is hypercyclic, (ii) JG(vη) = ℝn for some vector vη given by the structure of G, (iii) G(vη) = ℝn. This answer a question raised by the author. Moreover we construct for any n = 2 an abelian semigroup G of GL(n, ℝ) generated by n + 1 diagonal matrices which is locally hypercyclic (or J-class) but not hypercyclic and such that JG(ek) = ℝn for every k = 1,…, n, where (e1,…, en) is the canonical basis of ℝn. This gives a negative answer to a question raised by Costakis and Manoussos

J-class abelian semigroups of matrices on ℝ<italic>n</italic>

Applied Mathematics and Nonlinear Sciences

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J-class abelian semigroups of matrices on ℝ

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We establish, for finitely generated abelian semigroups G of matrices on ℝn, and by using the extended limit sets (the J-sets), the following equivalence...

J-class abelian semigroups of matrices on ℝⁿ

Published Online: Dec 08, 2017

Page range: 519 - 528

Received: May 07, 2017

Accepted: Dec 08, 2017

DOI: https://doi.org/10.21042/AMNS.2017.2.00043

Keywords
Hypercyclic semigroup, locally hypercyclic, J-class operator, extended limit set, dense orbit, semigroup

© 2017 Habib Marzougui published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

J-class abelian semigroups of matrices on ℝn

Published Online: Dec 08, 2017

Page range: 519 - 528

Received: May 07, 2017

Accepted: Dec 08, 2017

DOI: https://doi.org/10.21042/AMNS.2017.2.00043

KeywordsHypercyclic semigroup, locally hypercyclic, J-class operator, extended limit set, dense orbit, semigroup

© 2017 Habib Marzougui published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

J-class abelian semigroups of matrices on ℝⁿ

Keywords
Hypercyclic semigroup, locally hypercyclic, J-class operator, extended limit set, dense orbit, semigroup