Polynomial automorphisms and hypercyclic operators on spaces of analytic functions

Abstract

We consider hypercyclic composition operators on \(H({\mathbb{C}}^{n})\) which can be obtained from the translation operator using polynomial automorphisms of \({\mathbb{C}}^{n}\) . In particular we show that if CS is a hypercyclic operator for an affine automorphism S on \(H({\mathbb{C}}^{n})\) , then \(S = \Theta \circ (I + b) \circ \Theta ^{-1} + a\) for some polynomial automorphism Θ and vectors a and b, where I is the identity operator. Finally, we prove the hypercyclicity of “symmetric translations” on a space of symmetric analytic functions on l1.