A. Manoussos | Searcharxiv

arXiv2011

Properness, Cauchy-indivisibility and the Weil completion of a group of isometries

In this paper we introduce a new class of metric actions on separable (not necessarily connected) metric spaces called "Cauchy-indivisible" actions. This new class coincides with that of proper actions on locally compact metric spaces and, as examples show, it may be different in general. The concept of "Cauchy-indivisibility" follows a more general research direction proposal in which we investigate the impact of basic notions in substantial results, like the impact of local compactness and connectivity in the theory of proper transformation groups. In order to provide some basic theory for this new class of actions we embed a "Cauchy-indivisible" action of a group of isometries of a separable metric space in a proper action of a semigroup in the completion of the underlying space. We show that, in case this subgroup is a group, the original group has a "Weil completion" and vice versa. Finally, in order to establish further connections between "Cauchy-indivisible" actions on separable metric spaces and proper actions on locally compact metric spaces we investigate the relation between "Borel sections" for "Cauchy-indivisible" actions and "fundamental sets" for proper actions. Some open questions are added.

math.GN↗

arXiv2010

Dynamics of tuples of complex upper triangular Toeplitz matrices

Based on a question raised by N. Feldman we discuss the dynamics of tuples of upper triangular Toeplitz matrices over $\mathbb{C}$. Some open problems are posed.

math.FA↗

arXiv2009

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

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 $\mathbb{R}$ or $\mathbb{C}$, 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 $\mathbb{R}^n$ is $n+1$, thus complementing a recent result due to Feldman.

math.FA↗