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

Abstract

In this paper we extend the notion of a locally hypercyclic operator to that of a locally hypercyclic tuple of operators. We then show that the class of hypercyclic tuples of operators forms a proper subclass to that of locally hypercyclic tuples of operators. What is rather remarkable is that in every finite dimensional vector space over R or C, a pair of commuting matrices exists which forms a locally hypercyclic, non-hypercyclic tuple. This comes in direct contrast to the case of hypercyclic tuples where the minimal number of matrices required for hypercyclicity is related to the dimension of the vector space. In this direction we prove that the minimal number of diagonal matrices required to form a hypercyclic tuple on R n is n + 1, thus complementing a recent result due to Feldman.

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

“…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) = C n (Theorem 1.2). Secondly, we answer negatively (Theorem 1.5) the following question raised by Costakis and Manoussos in [5]: is it true that a locally hypercyclic abelian semigroup G generated by matrices A 1 , . .…”

“…We refer the reader to the recent book [3] and [7] for a thorough account on hypercyclicity. In [5], Costakis and Manoussos introduced the concept of extended limit set to G: Suppose that G is generated by p matrices A 1 , . .…”

Topologically transitive semigroup actions of real linear fractional transformations

“…On the other hand, no n diagonal matrices can generate a hypercyclic semigroup [10]. The action of the group of diagonal matrices on R n is not weakly topologically mixing (see Problem 3 in Section 6 in relation to this observation).…”

Chaotic behavior of discrete-time linear inclusion dynamical systems

“…It is often very unpredictable under the "generic" switching laws in Σ + K . In some literature, for example [28,18,19], the complexity of System (1.1) is described by the existence of a dense trajectory (x n (x 0 , σ)) n≥1 in R d (i.e. {x n (x 0 , σ) | n = 1, 2, .…”