# Unilateral weighted shifts on $\ell^2$

## TL;DR

This paper explores the complexity of vectors in  that are hypercyclic for all weighted shifts in a certain set, showing how to construct such sets with arbitrarily complicated hypercyclic vectors.

## Contribution

It demonstrates how to construct sets of weights in ^ that produce arbitrarily complex sets of hypercyclic vectors in .

## Key findings

- The set of hypercyclic vectors can be made arbitrarily complicated.
- Construction of weight sets W with prescribed hypercyclic vector complexity.
- Hypercyclicity behavior can be controlled via the choice of W.

## Abstract

Given a unilateral shift $B_w$ (determined by a bounded sequence $w$), a sequence $x \in \ell^2$ is "hypercyclic" for $w$ iff the forward iterates of $x$ under $B_w$ are dense in $\ell^2$. We show that it is possible to make the set of $x \in \ell^2$ which are simultaneously hypercyclic for all $w$ in a fixed $W \subseteq \ell^\infty$ arbitrarily complicated by choosing $W$ appropriately.