Hypercyclic abelian semigroup of matrices on Cn and Rn and k-transitivity (k ≥ 2)

Abstract

We prove that the minimal number of matrices on C n required to form a hypercyclic abelian semigroup on C n is n + 1. We also prove that the action of any abelian semigroup finitely generated by matrices on C n or R n is never k-transitive for k ≥ 2. These answer questions raised by Feldman and Javaheri.

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

“…Recently there has been done much research about this subject. We mention only [15], [3], [8], [9], [4], [5], [10] for the abelian case and [2], [13] for the non-abelian case. Our methods differ from those in the previously mentioned papers since we use methods from the theories of algebraic groups and algebraic actions.…”

Abelian Semigroups of Matrices on ℂnand Hypercyclicity

“…Secondly, we prove that the minimal number of matrices required to form a hypercyclic abelian semigroup in K η (C), having a normal form of length r (see the definition below), is exactly 2n − r + 1 (see Corollary 1.7). In particular, n + 1 is the minimal number of matrices on C n required to form a hypercyclic abelian semigroup on C n ; this was recently shown in [2], answering a question raised by Feldman in [8, § 6].…”

“…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.…”

线性算子动力系统的研究进展

“…Javaheri [60] 讨论了有限元生成的算子半群的超循环性质, 并给出了具体的例子, 即存在 n × n 矩阵 A 和 B, 使得对于半群映射 ⟨A, B⟩ 在任意列向量下的轨道在 K n 中是稠密的. 而且, Ayadi [61] 证明了, [62] 提出. [63] 证明了存在超循环的对角矩阵 n + 1 元组, 但是 不存在超循环的对角矩阵 n 元组.…”