Disjoint hypercyclicity, Sidon sets and weakly mixing operators

TL;DR

This paper explores the relationship between disjoint hypercyclicity, Sidon sets, and weakly mixing operators, establishing conditions under which hypercyclicity implies weak mixing and constructing specific operators with particular hypercyclic properties.

Contribution

It characterizes when disjoint hypercyclicity implies weak mixing based on Sidon set properties and constructs an example of a non-weakly mixing operator with hypercyclic direct sums.

Findings

A finite set J with J∪{0} not Sidon implies disjoint hypercyclicity leads to weak mixing.

Existence of a non-weakly mixing operator T with hypercyclic direct sums T⊕T²⊕...⊕Tⁿ for all n.

Abstract

We prove that a finite set of natural numbers $J$ satisfies that $J\cup\{0\}$ is not Sidon if and only if for any operator $T$, the disjoint hypercyclicity of $\{T^j:j\in J\}$ implies that $T$ is weakly mixing. As an application we show the existence of a non weakly mixing operator $T$ such that $T\oplus T^2\ldots \oplus T^n$ is hypercyclic for every $n$.