Dynamics of composition operators on spaces of holomorphic functions on plane domains

We study the dynamic behaviour of (weighted) composition operators on the space of holomorphic functions on a plane domain. Any such operator is hypercyclic if and only if it is topologically mixing, and when the symbol is automorphic, such an operator is supercyclic if and only if it is mixing. When the domain is a punctured plane, a composition operator is supercyclic if and only if it satisfies the Frequent Hypercyclity Criterion, and when the domain is conformally equivalent to a punctured disc, such an operator is hypercyclic if and only if it satisfies the Frequent Hypercyclicity Criterion. When the domain is finitely connected and either conformally equivalent to an annulus or having two or more holes, no weighted composition operator can be supercyclic.