Chaos in convolution operators on the space of entire functions of infinitely many complex variables

TL;DR

This paper investigates the dynamics of convolution operators on the space of entire functions with infinitely many variables, revealing they are not cyclic but exhibit Li--Yorke chaos, contrasting with finite-variable cases.

Contribution

It demonstrates that convolution operators on infinite-variable entire functions are not cyclic or supercyclic but are Li--Yorke chaotic, extending chaos theory to non-metrizable spaces.

Findings

No convolution operator is cyclic or n-supercyclic on al(^\u221E)

Every nontrivial convolution operator is Li--Yorke chaotic

Contrasts with finite-variable hypercyclicity results

Abstract

A classical result of Godefroy and Shapiro states that every nontrivial convolution operator on the space $\mathcal{H}(\mathbb{C}^n)$ of entire functions of several complex variables is hypercyclic. In sharp contrast with this result F\'avaro and Mujica show that no translation operator on the space $\mathcal{H}(\mathbb{C}^\mathbb{N})$ of entire functions of infinitely many complex variables is hypercyclic. In this work we study the linear dynamics of convolution operators on $\mathcal{H}(\mathbb{C}^\mathbb{N})$. First we show that no convolution operator on $\mathcal{H}(\mathbb{C}^\mathbb{N})$ is neither cyclic nor $n$-supercyclic for any positive integer $n$. After we study the notion of Li--Yorke chaos in non-metrizable topological vector spaces and we show that every nontrivial convolution operator on $\mathcal{H}(\mathbb{C}^\mathbb{N})$ is Li--Yorke chaotic.