Denjoy-Wolff like set for rational semigroups

In this paper, we present a new technique for studying the dynamics of a finitely generated rational semigroup. Such a semigroup can be associated naturally to a certain holomorphic correspondence on [Formula: see text]. Results on the iterative dynamics of such a correspondence can now be applied to the study of the rational semigroup. We focus on an invariant measure for the aforementioned correspondence — known as the equilibrium measure. This confers some advantages over many of the known techniques for studying the dynamics of rational semigroups. We use the equilibrium measure to analyze the distribution of repelling fixed points of a finitely generated rational semigroup, and to derive a sharp bound for the Hausdorff dimension of the Julia set of such a semigroup.

Easy multiplications II. Extensions of rational semigroups

This paper is based on the author's previous work (Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products. Ergod. Th. & Dynam. Sys. 21 (2001), 563–603). We consider fiber-preserving complex dynamics on fiber bundles whose fibers are Riemann spheres and whose base spaces are compact metric spaces. In this context, without any assumption on (semi-)hyperbolicity, we show that the fiberwise Julia sets are uniformly perfect. From this result, we show that, for any semigroup G generated by a compact family of rational maps on the Riemann sphere ) that is semi-hyperbolic, except at perhaps finitely many points in the Julia set, and which satisfies the open set condition, is either porous or equal to the closure of the open set. Furthermore, we derive an upper estimate of the Hausdorff dimension of the Julia set.

A correction to the proof of a lemma in ‘Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products’