Ising models, julia sets, and similarity of the maximal entropy measures

We study the phase transition of Ising models on diamondlike hierarchical lattices. Following an idea of Lee and Yang, one can make an analytic continuation of free energy of this model to the complex temperature plane. It is known that the Migdal-Kadanoff renormalization group of this model is a rational endomorphism (denoted by f) of Ĉ and that the singularities of the free energy lie on the Julia set J(f). The aim of this paper is to prove that the free energy can be represented as the logarithmic potential of the maximal entropy measure on J(f). Moreover, using this representation, we can show a close relationship between the critical exponent and local similarity of this measure.