We introduce the Plaque Topology on the inverse limit of a branched covering self-map of a Riemann surface of a finite degree greater than one. We present the notions of regular and irregular points in the setting of this Plaque Inverse Limit and study its local topological properties at the irregular points. We construct a certain Boolean algebra and a certain sigma-lattice, derived from it, and use them to compute local topological invariants of the Plaque Inverse Limit. Finally, we obtain several results interrelating the dynamics of the forward iterations of the self-map and the topology of the Plaque Inverse Limit.

There is a classical extension, of Möbius automorphisms of the Riemann sphere into isometries of the hyperbolic space H 3 , which is called the Poincaré extension. In this paper, we construct extensions of rational maps on the Riemann sphere over endomorphisms of H 3 exploiting the fact that any holomorphic covering between Riemann surfaces is Möbius for a suitable choice of coordinates. We show that these extensions define conformally natural homomorphisms on suitable subsemigroups of the semigroup of Blaschke maps. We extend the complex multiplication to a product in H 3 that allows to construct a visual extension of any given rational map.

We use semigroup theory to describe the group of automorphisms of some semigroups of interest in holomorphic dynamical systems. We show, with some examples, that representation theory of semigroups is related to usual constructions in holomorphic dynamics. The main tool for our discussion is a theorem due to Schreier. We extend this theorem, and our results in semigroups, to the setting of correspondences and holomorphic correspondences.

In this note, we show that the regular part of the natural extension (in the sense of Lyubich and Minsky 1997, J. Diff. Geom., 49, pp. 17 -94) of quadratic map f ðzÞ ¼ e 2piu z þ z 2 with irrational u of bounded type has only parabolic leaves except the invariant lift of the Siegel disc. We also show that though the natural extension of a rational function with a Cremer fixed point has a continuum of irregular points, it cannot supply enough singularity to apply the Gross star theorem to find hyperbolic leaves.

Abstract. If R is a rational map, the main result is a uniformization theorem for the space of decompositions of the iterates of R. Secondly, we show that Fatou conjecture holds for decomposable rational maps.