Fatou, Julia, and escaping sets in holomorphic (sub)semigroup dynamics

Abstract

We investigate under what conditions the Fatou, Julia, and escaping sets of a transcendental semigroup arerespectively equal to the Fatou, Julia, and escaping sets of their subsemigroups. We define the partial fundamental setand fundamental set of a holomorphic semigroup, and on the basis of these sets, we prove that the Fatou and escapingsets of a transcendental semigroup S are nonempty.