Completely Invariant Fatou, Julia, and Escaping Sets of Transcendental Semigroups

Abstract

In holomorphic semigroup dynamics, Fatou and escaping sets are, in general, forward invariant and Julia sets are backward invariant. Therefore, some fundamental results of the holomorphic dynamics of a single holomorphic function cannot be generalized and preserved for holomorphic semigroup dynamics. In this paper, we define completely invariant Julia, Fatou, and escaping sets of transcendental semigroups, and we see how far the results of the holomorphic dynamics of a single transcendental entire function can be preserved and generalized for transcendental semigroup dynamics.