  We consider a class of dynamical systems, which we call weakly coarse
expanding, which is a generalization to the postcritically infinite case of
expanding Thurston maps as discussed by Bonk-Meyer and is closely related to
coarse expanding conformal systems as defined by Haissinsky-Pilgrim. We prove
existence and uniqueness of equilibrium states for a wide class of potentials,
as well as statistical laws such as a central limit theorem, law of iterated
logarithm, exponential decay of correlations and a large deviation principle.
Further, if the system is defined on the 2-sphere, we prove all such results
even in presence of periodic (repelling) branch points.
