The interplay between recurrence and hypercyclicity in dissipative contexts

TL;DR

This paper explores the relationship between recurrence and hypercyclicity in dissipative composition operators, demonstrating their equivalence and addressing open questions in linear dynamics.

Contribution

It establishes the equivalence of recurrence and hypercyclicity notions in dissipative composition operators, improving previous results and answering open questions.

Findings

Recurrence and hypercyclicity are equivalent in dissipative composition operators.

The paper improves existing theorems related to these notions.

It provides solutions to previously open questions in the field.

Abstract

Motivated by recent investigations \cite{Costakis, Bonilla} on the notion of recurrence in linear dynamics, we deepen into the notions of recurrence and frequent recurrence in the setting of dissipative composition operators with bounded distortion, a class of linear operators which includes backward shifts. Among other results, we show that these two notions are, actually, equivalent to those of hypercyclicity and frequent hypercyclicity, respectively. More particularly, we improve \cite[Theorem 5.2]{Bonilla} and, also, provide an answer to an open question posed in \cite[Question 5.3]{Bonilla}.