We give a sufficient condition for two operators to be disjointly frequently hypercyclic. apply this criterion composition acting on H(D) or the Hardy space H2(D). simplify result disjoint frequent hypercyclicity of pseudo shifts recent paper Martin et al. and we exhibit hypercyclic weighted shifts.

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

We provide with criteria for a family of sequences operators to share frequently universal vector. These are inspired by the classical Frequent Hypercyclicity Criterion and recent criterion due Grivaux, Matheron Menet where periodic points play central role. As an application, we obtain any operator T in specific class acting on separable Banach space, necessary sufficient condition subset ? co...

We study positive shadowing and chain recurrence in the context of linear operators acting on Banach spaces or even normed vector spaces. show that for there is only one recurrent set, this set a closed invariant subspace. prove every transitive dynamical system with property frequently hypercyclic and, as corollary, we obtain hypercyclic.

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...