Multiple recurrence and hypercyclicity

Abstract

We study multiply recurrent and hypercyclic operators as a special case of F -hypercyclicity, for F being the family of subsets of the natural numbers containing arbitrarily long arithmetic progressions. We prove that every separable infinite dimensional Banach space supports hypercyclic multiply recurrent operators, we characterize those operators which are weakly mixing and multiply recurrent and we show that there are operators that are multiply recurrent and hypercyclic but not weakly mixing.

Disjoint hypercyclicity, Sidon sets and weakly mixing operators

“…However, operators can be thickly transitive and syndetically transitive. For more on F -hypercyclicity see [6,7,10,11,12,13,18].…”