Hypercyclic algebras for convolution and composition operators

TL;DR

This paper demonstrates the existence of hypercyclic algebras for certain convolution and differentiation operators on spaces of entire functions, providing new examples beyond polynomial-induced operators and contrasting with non-supporting cases.

Contribution

It offers an alternative proof for hypercyclic algebras supported by the differentiation operator and extends this to convolution operators not induced by polynomials.

Findings

Differentiation operator supports hypercyclic algebra.

Certain convolution operators like cos(D) and e^D-aI support hypercyclic algebras.

Weighted composition operators do not support supercyclic algebras.

Abstract

We provide an alternative proof to those by Shkarin and by Bayart and Matheron that the operator $D$ of complex differentiation supports a hypercyclic algebra on the space of entire functions. In particular we obtain hypercyclic algebras for many convolution operators not induced by polynomials, such as $\mbox{cos}(D)$, $De^D$, or $e^D-aI$, where $0<a\le 1$. In contrast, weighted composition operators on function algebras of analytic functions on a plane domain fail to support supercyclic algebras. Non-trivial translations on the space of complex-valued, smooth functions on the real line do support hypercyclic algebras.