Dynamics of tuples of matrices

Abstract

Abstract. In this article we answer a question raised by N. Feldman in 2008 concerning the dynamics of tuples of operators on R n . In particular, we prove that for every positive integer n ≥ 2 there exist n-tuples (A 1 , A 2 , . . . , A n ) of n × n matrices over R such that (A 1 , A 2 , . . . , A n ) is hypercyclic. We also establish related results for tuples of 2 × 2 matrices over R or C being in Jordan form.

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

“…, s n ∈ N ∪ {0}} is dense in (R + ) n . An easy argument (see for example the proof of lemma 2.6 in [3]) shows that the set…”

Semigroups of matrices with dense orbits

“…In [5], Feldman initiated the study of hypercyclic semigroups of linear operators in the finite-dimensional case and proved that, in dimension n, there exists a hypercyclic semigroup generated by n + 1 diagonalizable matrices (Costakis et al [3] proved that it is not possible to reduce the number of generators to less than n + 1). If one removes the diagonalizability condition, it is shown by Costakis et al [4] that one can find a hypercyclic abelian semigroup of n matrices in dimension n. It is then natural to consider the non-commuting case.…”

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, .…”