Jordan domain and Fatou set concerning diamond-like hierarchical Potts models**The research was supported by the National Natural Science Foundation of China (Grant No 10625107) and the Program for New Century Excellent Talents in University (Grant No 04-0490).

Abstract

For the Potts models on the diamond-like hierarchical lattice, the domains of the complex phases are indeed the Fatou components of a family of rational maps. In this paper, we deal with the relationships between this family of Fatou components and the Jordan domains and describe the topological structures of this family of Fatou components completely.