  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.
