On the topology of the inverse limit of a branched covering over a Riemann surface

TL;DR

This paper introduces the Plaque Topology for inverse limits of branched coverings on Riemann surfaces, analyzing local topological properties and their relation to the dynamics of the map.

Contribution

It defines the Plaque Topology, studies local properties at irregular points, and links the topology of the inverse limit to the dynamics of the self-map.

Findings

Characterization of regular and irregular points

Construction of Boolean algebra and sigma-lattice for invariants

Results relating dynamics to inverse limit topology

Abstract

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 certain Boolean Algebra and 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.