Compositionally Universal Entire Functions on the Plane and the Punctured Plane

Abstract

Necessary/sufficient conditions for a sequence of automorphisms of the complex plane to generate a sequence of composition operators that is universal on the punctured plane are provided. As a consequence, it is derived that only for translations and rotation-dilations there can be entire functions whose orbits present universality. Boundedness of these functions on unbounded sets as well as frequent universality on the whole plane are also analyzed.

Universal sequences of composition operators

“…The topic has been extensively continued in various directions later on, see for e.g. [1,3,4,5,6,13,14,15,17,22].…”

“…n (I m ) ∪ φ n (U) ∪ L has connected complement. By(1) andLemma 4.11, it is enough to prove thatl m=1 φ n (I m ) ∪ φ n (U)has connected complement. Let J 0 = e iα , e iδ 1 , J l = e iδ 2l , e iβ and J m = e iδ 2m , e iδ 2m+1 , m = 1, .…”