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A classical result of Godefroy and Shapiro states that every nontrivialconvolution operator on the space $\mathcal{H}(\mathbb{C}^n)$ of entirefunctions of several complex variables is hypercyclic. In sharp contrast withthis result F\'avaro and Mujica show that no translation operator on the space$\mathcal{H}(\mathbb{C}^\mathbb{N})$ of entire functions of infinitely manycomplex variables is hypercyclic. In this work we study the linear dynamics ofconvolution operators on $\mathcal{H}(\mathbb{C}^\mathbb{N})$. First we showthat no convolution operator on $\mathcal{H}(\mathbb{C}^\mathbb{N})$ is neithercyclic nor $n$-supercyclic for any positive integer $n$. After we study thenotion of Li--Yorke chaos in non-metrizable topological vector spaces and weshow that every nontrivial convolution operator on$\mathcal{H}(\mathbb{C}^\mathbb{N})$ is Li--Yorke chaotic.

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