Semigroups of matrices with dense orbits

Abstract

We give examples of n × n matrices A and B over the filed K = R or C such that for almost every column vector x ∈ K n , the orbit of x under the action of the semigroup generated by A and B is dense in K n .

On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple

“…We stress that all the tuples considered in this work consist of commuting matrices/operators. Recently, in [11] Javaheri deals with the non-commutative case. In particular, he shows that for every positive integer n ≥ 2 there exist non-commuting linear maps A, B : R n → R n so that for every vector x = (x 1 , x 2 , .…”

Topologically transitive semigroup actions of real linear fractional transformations

“…On the other hand, by the results in [19], for every n there exist two matrices that generate a topologically transitive semigroup action on R n . However, the construction in [19] does not give a weakly topologically mixing semigroup action. In this direction, the following question arises.…”

“…The general theory of hypercyclicity was initiated by Rolewicz [21] who introduced the first examples of hypercyclic operators in a Banach space. Rolewicz also showed that no finite-dimensional Banach space admits a hypercyclic operator (however, one can construct a pair of linear operators that generate a hypercyclic semigroup in the sense defined in the sequel; see [19]). On the other hand, Ansari et al [1,4,6] proved that every separable infinitedimensional Fréchet space admits a hypercyclic operator.…”

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 .…”