We show that there exists an invertible frequently hypercyclic operator on $\ell^1(\mathbb{N})$ whose inverse is not hypercyclic.

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 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 solve a number of questions pertaining to the dynamics linear operators on Hilbert spaces, sometimes by using Baire category arguments and by constructing explicit examples. In particular, we prove following results. (i) A typical hypercyclic operator is not topologically mixing, has no eigenvalues admits non-trivial invariant measure, but densely distributionally chaotic. (ii) {upp...

In this paper we characterize hypercyclic sequences of weighted translation operators on an Orlicz space in the context locally compact hypergroups.