Hypercyclic abelian semigroup of matrices on Cn and Rn and k-transitivity (k ≥ 2)

Authors

  • Adlene Ayadi University of Gafsa

DOI:

https://doi.org/10.4995/agt.2011.1699

Keywords:

Hypercyclic, Tuple of matrices, Semigroup, Subgroup, Dense orbit, Transitive, Semigroup action

Abstract

We prove that the minimal number of matrices on Cn required to forma hypercyclic abelian semigroup on Cn is n+1. We also prove that theaction of any abelian semigroup finitely generated by matrices on Cnor Rn is never k-transitive for k 2. These answer questions raised byFeldman and Javaheri.

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Author Biography

Adlene Ayadi, University of Gafsa

Department of Mathematics, Faculty of Science of Gafsa, 2112, Gafsa, Tunisia.

References

A. Ayadi and H. Marzougui, Dynamic of Abelian subgroups of GL(n, C): a structure Theorem, Geom. Dedicata 116 (2005), 111–127. http://dx.doi.org/10.1007/s10711-005-9007-2

A. Ayadi and H. Marzougui, Dense orbits for abelian subgroups of GL(n, C), Foliations 2005: World Scientific, Hackensack, NJ (2006), 47–69.

F. Bayart and E. Matheron, Dynamics of Linear Operators, Cambridge Tracts in Math., 179, Cambridge University Press, 2009. http://dx.doi.org/10.1017/CBO9780511581113

G. Costakis, D. Hadjiloucas and A. Manoussos, Dynamics of tuples of matrices, Proc. Amer. Math. Soc. 137, no. 3 (2009), 1025–1034. http://dx.doi.org/10.1090/S0002-9939-08-09717-7

G. Costakis, D. Hadjiloucas and A. Manoussos, On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple, J. Math. Anal. Appl. 365 (2010), 229–237. http://dx.doi.org/10.1016/j.jmaa.2009.10.020

N. S. Feldman, Hypercyclic tuples of operators and somewhere dense orbits, J. Math. Anal. Appl. 346 (2008), 82–98. http://dx.doi.org/10.1016/j.jmaa.2008.04.027

M. Javaheri, Topologically transitive semigroup actions of real linear fractional transformations , J. Math. Anal. Appl. 368 (2010), 587–603. http://dx.doi.org/10.1016/j.jmaa.2010.03.028

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How to Cite

[1]
A. Ayadi, “Hypercyclic abelian semigroup of matrices on Cn and Rn and k-transitivity (k ≥ 2)”, Appl. Gen. Topol., vol. 12, no. 1, pp. 35–39, Apr. 2011.

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