The process of finding f ' for an entire function f has infinite topological entropy

Abstract

Let (H (C), p) be the metric space of all entire functions f where the metric p induces the topology of uniform convergence on compact subsets of the complex plane. Let D : H (C) -> H (C) be the linear mapping that assigns to each f its derivative, D (f) = f'. We show in this note that there exists a compact subset of H (C), say K, that is invariant under D, and D restricted to K has infinite topological entropy.