Multiple recurrence and hypercyclicity

TL;DR

This paper investigates the properties of multiply recurrent and hypercyclic operators within the framework of $ ext{F}$-hypercyclicity, providing characterizations and examples of operators with specific recurrence and mixing behaviors.

Contribution

It introduces a new perspective on hypercyclicity by studying multiply recurrent operators as a special case of $ ext{F}$-hypercyclicity and characterizes their properties.

Findings

Characterized operators that are weakly mixing and multiply recurrent.

Proved properties of hypercyclic multiply recurrent operators.

Constructed examples of operators that are multiply recurrent and hypercyclic but not weakly mixing.

Abstract

We study multiply recurrent and hypercyclic operators as a special case of $\mathcal F$-hypercyclicity, where $\mathcal F$ is the family of subsets of the natural numbers containing arbitrarily long arithmetic progressions. We prove several properties of 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.