Hypercyclic algebras for weighted shifts on trees

TL;DR

This paper investigates the conditions under which hypercyclic vectors form algebras for weighted backward shifts on sequence spaces over directed trees, revealing when such algebras exist or not.

Contribution

It provides necessary and sufficient conditions for hypercyclic algebras on various sequence spaces over directed trees, including new examples and non-existence results.

Findings

Hypercyclic algebras exist on $c_0(V)$ and $(V)$ when non-empty.

Characterization of hypercyclic algebras on $^p(V)$ for $1<p<+$.

Examples of hypercyclic operators without hypercyclic algebras.

Abstract

We study the existence of algebras of hypercyclic vectors for weighted backward shifts on sequence spaces of directed trees with the coordinatewise product. When $V$ is a rooted directed tree, we show the set of hypercyclic vectors of any backward weighted shift operator on the space $c_0(V)$ or $\ell^1(V)$ is algebrable whenever it is not empty. We provide necessary and sufficient conditions for the existence of these structures on $\ell^p(V), 1<p<+\infty$. Examples of hypercyclic operators not having a hypercyclic algebra are found. We also study the existence of mixing and non-mixing backward weighted shift operators on any rooted directed tree, with or without hypercyclic algebras. The case of unrooted trees is also studied.