Topological generators of abelian Lie groups and hypercyclic finitely generated abelian semigroups of matrices

Abstract

In this paper we bring together results about the density of subsemigroups of abelian Lie groups, the minimal number of topological generators of abelian Lie groups and a result about actions of algebraic groups. We find the minimal number of generators of a finitely generated abelian semigroup or group of matrices with a dense or a somewhere dense orbit by computing the minimal number of generators of a dense subsemigroup (or subgroup) of the connected component of the identity of its Zariski closure.

Abelian Semigroups of Matrices on ℂnand Hypercyclicity

“…They have, independently, proved similar results to Corollaries 1.8 and 1.9. The methods of proof in [1,12] and in this paper are quite different and have different consequences.…”

“…Shkarin [12] and Abels and Manoussos [1] considered the same topic, in particular the minimal number of generators of a finitely abelian hypercyclic semigroup of matrices on C n and R n . They have, independently, proved similar results to Corollaries 1.8 and 1.9.…”

“…Recently, there has been much research around this subject. We mention, in particular, [1][2][3][5][6][7][8]12] for the abelian case and [10] for the non-abelian case. Feldman showed in [8] that in C n there exists a hypercyclic semigroup generated by an (n + 1)-tuple of diagonal matrices on C n , and that no semigroup generated by an n-tuple of diagonalizable matrices on C n or R n can be hypercyclic.…”

“…. , r, (e (k) η ) j = 0 ∈ C nj if j = k, e η,k if j = k.An equivalent formulation is e(1) η = e 1 , . .…”

Approximately nearly controllability of discrete-time bilinear control systems with control input characteristic

“…Hence we shall prove our main result by studying the set of all hypercyclic points of S U . (1) It is worthy of mentioning that the hypercyclicity of a communicative semigroup generated by finite number of matrices has been extensively studied, for instance, see [13,14,15,16,17] and references therein. In particular, Costakis and Perissis recently proved that there exists a communicative matrix semigroup generated by n + 1 real matrices of Jordan form is hypercylic on R n .…”