J-class abelian semigroups of matrices on ℝⁿ

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) J_G(v_η) = ℝⁿ for some vector v_η given by the structure of G, (iii) G(v_η) = 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 J_G(e_k) = ℝⁿ for every k = 1,…, n, where (e₁,…, e_n) is the canonical basis of ℝⁿ. This gives a negative answer to a question raised by Costakis and Manoussos