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In this paper, we bring together results about the existence of a somewhere dense (resp. dense) orbit and the minimal number of generators for abelian semigroups of matrices on Rn. We solve the problem of determining the minimal number of matrices in normal form over R which form a hypercyclic abelian semigroup on Rn. In particular, we show that no abelian semigroup generated by [n+12] matrices on...
We give a characterization of hypercyclic finitely generated abelian semigroups of matrices on $$\mathbb{C }^{n}$$ using the extended limit sets (the J-sets). Moreover we construct for any $$n\ge 2$$ an abelian semigroup $$G$$ of GL $$(n, \mathbb{C })$$ generated by $$n+1$$ diagonal matrices which is locally hypercyclic but not hypercyclic and such that J $$_{G}(e_{k}) =...
We give a complete characterization of a supercyclic abelian semigroup of matrices on $$\mathbb {C}^{n}$$ C n . For finitely generated semigroups, this characterization is explicit and it is used to determine the minimal number of matrices in normal form over $$\mathbb {C}$$ C that form a supercyclic abelian semigroup on $${\mathbb {C}}^{n}$$ C n . In particular, no...
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