Power bounded composition operators on spaces of meromorphic functions

We study composition operators with holomorphic symbols defined on spaces of meromorphic functions, when endowed with their natural locally convex topology. First, we show that such operators are well-defined, continuous and never compact. Then, we study the dynamics and prove that a composition operator is power bounded or mean ergodic if and only if the symbol is a nilpotent element in the group of automorphisms.