On the dynamics of composite entire functions

Abstract

Letf and g be nonlinear entire functions. The relations between the dynamics of f⊗g and g⊗f are discussed. Denote byℐ (·) and F(·) the Julia and Fatou sets. It is proved that if z∈C, then z∈ℐ8464 (f⊗g) if and only if g(z)∈ℐ8464 (g⊗f); if U is a component of F(f○g) and V is the component of F(g○g) that contains g(U), then U is wandering if and only if V is wandering; if U is periodic, then so is V and moreover, V is of the same type according to the classification of periodic components as U. These results are used to show that certain new classes of entire functions do not have wandering domains.