Common hypercyclic vectors and dimension of the parameter set

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 $\lambda \in \Lambda \mapsto T_{\lambda }^{n}x$, $x\in X$, $n\in \mathbb{N}$.