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.