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