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Abelian Semigroups of Matrices on ℂn and Hypercyclicity

Published online by Cambridge University Press:  05 September 2013

Adlene Ayadi  and
Habib Marzougui
Adlene Ayadi
Affiliation:
Department of Mathematics, Faculty of Science of Gafsa, University of Gafsa, Gafsa 2112, Tunisia, (adlenesoo@yahoo.fr)
Habib Marzougui
Affiliation:
Department of Mathematics, Faculty of Science of Bizerte, University of Carthage, Zarzouna 7021, Tunisia, (habib.marzougui@fsb.rnu.tn; hmarzoug@ictp.it)
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Abstract

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We give a complete characterization of a hypercyclic abelian semigroup of matrices on ℂn. For finitely generated semigroups, this characterization is explicit and it is used to determine the minimal number of matrices in normal form over ℂ that form a hypercyclic abelian semigroup on ℂn. In particular, we show that no abelian semigroup generated by n matrices on ℂn can be hypercyclic.

Keywords

hypercyclicmatricesorbitdense orbitsemigroupabelianPrimary 37C8547A16
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 2 , June 2014 , pp. 323 - 338
Copyright
Copyright © Edinburgh Mathematical Society 2014 

References

1.Abels, H. and Manoussos, A., Topological generators of abelian Lie groups and hyper-cyclic finitely generated abelian semigroups of matrices, Adv. Math. 229 (2012), 1862–1872.Google Scholar
2.Ayadi, A., Hypercyclic abelian semigroup of matrices on ℂn and ℝn and fc-transitivity (k ≥ 2), Appl. Gen. Topol. 12 (2011), 35–39.Google Scholar
3.Ayadi, A. and Marzougui, H., Dense orbits for abelian subgroups of GL(n, ℂ), in Foliations 2005 (ed. Walczak, P. G.), pp. 47–69 (World Scientific, 2006).CrossRefGoogle Scholar
4.Bayart, F. and Matheron, E., Dynamics of linear operators, Cambridge Tracts in Mathematics, Volume 179 (Cambridge University Press, 2009).Google Scholar
5.Costakis, G. and Parissis, I., Dynamics of tuples of matrices in Jordan form, Operators Matrices 7 (2013), 157–131.Google Scholar
6.Costakis, G., Hadjiloucas, D. and Manoussos, A., Dynamics of tuples of matrices, Proc. Am. Math. Soc. 137 (2009), 1025–1034.Google Scholar
7.Costakis, G., Hadjiloucas, D. and Manoussos, A., On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple, J. Math. Analysis Applic. 365 (2010), 229–237.Google Scholar
8.Feldman, N. S., Hypercyclic tuples of operators and somewhere dense orbits, J. Math. Analysis Applic. 346 (2008), 82–98.Google Scholar
9.Grosse-Herdmann, K. G. and Peris, A., Linear chaos, Universitext (Springer, 2011).CrossRefGoogle Scholar
10.Javaheri, M., Semigroups of matrices with dense orbits, Dynam. Sys. Int. J. 26 (2011), 235–243.Google Scholar
11.Rossmann, W., Lie groups: an introduction through linear groups, Oxford Graduate Texts in Mathematics, Volume 5 (Oxford University Press, 2002).CrossRefGoogle Scholar
12.Shkarin, S., Hypercyclic tuples of operator on ℂn and ℝn, Linear Multilinear Alg. 60(8) (2012), 885–896.Google Scholar
13.Suprunenko, D. A. and Tyshkevich, R. I., Commutative matrices (Academic, 1968).Google Scholar