Chaos and frequent hypercyclicity for weighted shifts

TL;DR

This paper explores the relationship between chaos and frequent hypercyclicity in weighted shift operators across various Banach and Fréchet sequence spaces, revealing conditions under which these properties coincide or differ.

Contribution

It generalizes the Bayart-Ruzsa theorem to Banach sequence spaces with unconditional bases and investigates the chaos-hypercyclicity relationship in Fréchet spaces, including K"othe and power series spaces.

Findings

Every frequently hypercyclic weighted shift on $H(bD)$ is chaotic.

There exists a non-chaotic frequently hypercyclic weighted shift on $H(bC)$.

The generalized theorem applies to all Banach sequence spaces with boundedly complete unconditional bases.

Abstract

Bayart and Ruzsa [Ergodic Theory Dynam. Systems 35 (2015)] have recently shown that every frequently hypercyclic weighted shift on $\ell^p$ is chaotic. This contrasts with an earlier result of Bayart and Grivaux [Proc. London Math. Soc. (3) 94 (2007)] who constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$. We first generalize the Bayart-Ruzsa theorem to all Banach sequence spaces in which the unit sequences are a boundedly complete unconditional basis. We then study the relationship between frequent hypercyclicity and chaos for weighted shifts on Fr\'echet sequence spaces, in particular on K\"othe sequence spaces, and then on the special class of power series spaces. We obtain, rather curiously, that every frequently hypercyclic weighted shift on $H(\mathbb{D})$ is chaotic, while $H(\mathbb{C})$ admits a non-chaotic frequently hypercyclic weighted shift.