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 $\lambda\in \Lambda\mapsto T\lambda^n x$, $x\in X$, $n\in\mathbb N$.