turkmath.org

A (continuos, linear) operator $T$ on a Frechet space $X$ is said to be hypercyclic provided it supports some vector $f$ in $X$ whose orbit $\{f, Tf, T^2f,...\}$ is dense in $X$. Such an $f$ is called a hypercyclic vector for $T$. A tuple $(T_1, T_2)$ of hypercyclic operators on $X$ is said to be disjoint-hypercyclic provided the direct sum $T_1\otimes T_2$ supports a hypercyclic vector of the form $(f, f)$ in $X\times X$. We contrast the dynamics of a single operator $T$ versus the disjoint dynamics of a tuple $(T_1, T_2)$. This includes joint work with J. Bes, A. Peris, R. Sanders, and S. Shkarin.