Absolutely Continuous Invariant Measure for Expanding Pick Maps of the Interval Except at a Marginal Fixed Point

It is an important problem in the theory of concrete dissipative dynamical systems to construct the relevant invariant measure. A rather complete theory has been developed for the case of uniformly hyperbolic dynamical systems following the initial ideas of Sinai Bowen and Ruelle. They proved an equivalence between questions about invariant measures and problems in statistical mechanics. In particular, the interesting invariant measures are Gibbs states of a one dimensional spin system.