turkmath.org

Let $T$ be a continuous and linear operator on a Banach space $X$. We say that $T$ is hypercyclic if there is some vector $x\in $X whose the orbit visits (infinitely often) each nonempty open set $U \subset X$. During the last decade, the researchers in Linear Dynamics have investigated the frequency of these visits and several variants of the notion of hypercyclicity have been introduced: the frequent hypercyclicity [1, 2], the U-frequent hypercyclicity [3] and the reiterative hypercyclicity [4]. The goal of this talk is to investigate the links between these different notions of hypercyclicity and their link with linear chaos. In particular, we answer one of the main current questions in Linear Dynamics by showing that there exists a chaotic operator on $\ell^1$ which is not frequently hypercyclic [5].