We study dynamical notions lying between $$\mathcal{U}$$ -frequent hypercyclicity and reiterative by investigating weighted upper densities the unweighted density Banach density. While chaos implies hypercyclicity, we show that does not imply with respect to any Moreover, if T is -frequently hypercyclic (resp. reiteratively hypercyclic) then n-fold product of still U-frequently this implicatio...

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