On dynamics of hyperbolic rational semigroups
The study of rational semigroups is a generalization of the study of Kleinian groups, iteration of rational functions and systems of contraction maps related to self-similar sets in \(\mathbb{C}\) in fractal geometry. In the framework of the study of rational semigroups, the author shows some basic results similar to those between Kleinian groups and iteration of rational functions. The author proves that if the hyperbolic rational semigroup is finitely generated and satisfies some natural conditions, the limit functions take their values on a postcritical set. Moreover, the author proves that when the generators of a finitely generated hyperbolic rational semigroup are perturbed, the hyperbolicity is kept and the Julia set depends continuously on the generators of the semigroup. Furthermore, if the finitely generated rational semigroup is hyperbolic and if the inverse images by the generators of the Julia set are mutually disjoint, then the Julia set moves by holomorphic motion.
scientific article; zbMATH DE number 1295314