Semigroups of transcendental entire functions and their dynamics (Q2358329)

A natural extension of dynamics of iterations of rational/transcendental functions is dynamics of semigroups of the functions. It is known that, in contrast to the iterations of a single function, the Fatou and Julia sets of a semigroup need not to be completely invariant. The authors extend here some results on iterations of transcendental functions to semigroups generated by transcendental functions. They give several sufficient conditions for the Julia set of a transcendental semigroup to be connected. For example, this is true for finitely generated semigroups with generators of bounded type (i.e., the critical points of their inverses are bounded). Abelian semigroups are considered as well. Finally, the authors study limit functions of transcendental semigroups.