Frequent hypercyclicity and piecewise syndetic recurrence sets

Abstract

Motivated by a question posed by Sophie Grivaux concerning the regularity of the orbits of frequently hypercylic operators, we show the following: for any operator $T$ on a separable metrizable and complete topological vector space $X$ which is both frequently hypercyclic and piecewise syndetic hypercyclic, the lower density and upper Banach density of the recurrence set $\{n\geq 1: T^n x\in U\}$ are different, for any hypercyclic vector $x\in X$ for $T$, and a certain collection of non-empty open sets $U\subseteq X$. As an immediate consequence we got a sufficient condition for a chaotic operator to be non frequently hypercyclic.