Common hypercyclic vectors and dimension of the parameter set

TL;DR

This paper explores the conditions under which a family of hypercyclic operators share a common hypercyclic vector, focusing on how the parameter set's dimension and regularity influence this existence.

Contribution

It provides new positive and negative results relating the dimension of the parameter set and the regularity of the operator family to the existence of common hypercyclic vectors.

Findings

Existence of common hypercyclic vectors depends on the dimension of the parameter set.

Regularity of the map from parameters to operators influences hypercyclicity.

Results include both positive and negative cases based on these properties.

Abstract

We investigate the existence of a common hypercyclic vector for a family $(T_\lambda)_{\lambda\in \Lambda}$ of hypercyclicoperators acting on the same Banach space $X$. We give positive and negative results involving the dimension of $\Lambda$ and the regularity of each map $\lambda\in \Lambda\mapsto T_\lambda^n x$, $x\in X$, $n\in\mathbb N$.