We consider a class of dynamical systems which we call weakly
coarse expanding,
which generalize to the postcritically infinite case expanding Thurston
maps as discussed by Bonk-Meyer
and are closely related to coarse expanding conformal systems as defined
by Haïssinsky-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.

This is a joint work with T. Das, F. Przytycki, G. Tiozzo, M. Urbański.