We study a family of renormalization transformations of generalized diamond
hierarchical Potts models through complex dynamical systems. We prove that the
Julia set (unstable set) of a renormalization transformation, when it is treated as a
complex dynamical system, is the set of complex singularities of the free energy in
statistical mechanics. We give a suﬃcient and necessary condition for the Julia sets
to be disconnected. Furthermore, we prove that all Fatou components (components
of the stable sets) of this family of renormalization transformations are Jordan
domains with at most one exception which is completely invariant. In view of
the problem in physics about the distribution of these complex singularities, we
prove here a new type of distribution: the set of these complex singularities in the
real temperature domain could contain an interval. Finally, we study the boundary
behavior of the first derivative and second derivative of the free energy on the Fatou
component containing the infinity. We also give an explicit value of the second order
critical exponent of the free energy for almost every boundary point.
Received by the editor June 25, 2011, and, in revised form, January 23, 2013.
Article electronically published on July 28, 2014.
2010 Mathematics Subject Classification. Primary 37F10, 37F45; Secondary: 82B20, 82B28.
Key words and phrases. Julia set, Fatou set, renormalization transformation, iterate, phase
The research was supported by the National Natural Science Foundation of China, the State Key
Development Program of Basic Research of China.
Aﬃliation at time of publication: School of Science and School of Computer Science, Beijing
University of Posts and Telecommunications, Beijing, 100876, People’s Republic of China; email:
2014 American Mathematical Society