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In this paper, we characterize the dynamic of every Abelian subgroup G of GL(n, K), K = R or C. We show that there exists a G-invariant, dense open set U in K n saturated by minimal orbits with K n − U a union of at most n G-invariant vector subspaces of K n of dimension n − 1 or n − 2 over K. As a consequence, G has height at most n and in particular it admits a minimal set in K n − {0}. (2000). 37C85. Mathematics Subject Classification

We prove that the minimal number of matrices on C n required to form a hypercyclic abelian semigroup on C n is n + 1. We also prove that the action of any abelian semigroup finitely generated by matrices on C n or R n is never k-transitive for k ≥ 2. These answer questions raised by Feldman and Javaheri.

We give a characterization of hypercyclic finitely generated abelian semigroups of matrices on C n using the extended limit sets (the J-sets). Moreover we construct for any n ≥ 2 an abelian semigroup G of GL(n, C) generated by n + 1 diagonal matrices which is locally hypercyclic but not hypercyclic and such that JG(e k ) = C n for every k = 1, . . . , n, where (e1, . . . , en) is the canonical basis of C n . This gives a negative answer to a question raised by Costakis and Manoussos.

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