J-class abelian semigroups of matrices on Rn

• Autores: Habib Marzougui
• Localización: Applied Mathematics and Nonlinear Sciences, ISSN-e 2444-8656, Vol. 2, Nº. 2, 2017, págs. 519-528
• Idioma: inglés
• DOI: 10.21042/amns.2017.2.00043
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• Resumen

  • 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.