Exponential type of hypercyclic entire functions

Abstract

In this paper the exponential type of hypercyclic entire functions with respect to a sequence¶ $ (\Phi_{n}(D)) $ of differential operators is considered, where every $ \Phi_n $ is an entire function of exponential type. We prove that under suitable conditions certain rates of growth are possible for hypercyclicity while others are not. In particular, our statements extend the negative part of a sharp result on growth of D-hypercyclic entire functions due to Grosse-Erdmann, and are related to a result by Chan and Shapiro about the existence of $ \Phi(D) $ -hypercyclic functions in certain Hilbert spaces of entire functions.