J-class abelian semigroups of matrices on $$\mathbb{C }^{n}$$ and hypercyclicity

Abstract

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.

J-class abelian semigroups of matrices on ℝ n

“…In the present work, we show that G is hypercyclic if and only if there exists a vector v in an open set V , defined according to the structure of G, such that J G (v) = R n (Theorem 1). This answer the question 1 raised by the author in [2]. Furthermore, we construct for every n ≥ 2, a locally hypercyclic abelian semigroup G generated by matrices A 1 , .…”

Hypercyclicity and subspace hypercyclicity in finite dimension

“…The first coordinate in (1) gives that m 1 + mα 1 = λ 1 x 1,1 . If x 1,1 = 0, then m = 0 and thus A α ∩ M ⊂ N n ∩ M which is clearly not dense in M , a contradiction.…”

Convex-cyclic matrices, convex-polynomial interpolation and invariant convex sets

“…Let v 1 and v 2 be eigenvectors for T * with eigenvalues λ and λ respectively. By (1) we know that λ / ∈ R, thus λ = λ. It follows that v 1 and v 2 are linearly independent.…”

“…Following Rezaei [12], we define an operator T to be convex-cyclic if there is a vector x ∈ X such that the convex-hull of the orbit of x under T is dense in X; that is if {p(T )x : p ∈ CP} is dense in X. Convex-cyclic operators were introduced by Rezaei [12] and have been studied in [3] and [11]. More generally the dynamics of matrices have been studied in [1], [5], [6] and in their references.…”