Frequent hypercyclicity and piecewise syndetic recurrence sets

TL;DR

This paper investigates the recurrence properties of frequently hypercyclic operators, showing that certain density measures differ for specific vectors and sets, and provides conditions under which chaotic operators are not frequently hypercyclic.

Contribution

It establishes a link between frequent hypercyclicity and piecewise syndetic recurrence, and offers a criterion to distinguish chaotic operators that are not frequently hypercyclic.

Findings

Lower density and upper Banach density differ for recurrence sets of certain vectors.

A sufficient condition is provided for chaotic operators to be non frequently hypercyclic.

The results answer a question posed by Sophie Grivaux regarding orbit regularity.

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.