2 INTRODUCTION

[GU], [KI], [LI], [MO1], [MO2]). However, properties of complex singularities of

antiferromagnetic models are much less well understood than those of ferromagnetic

models (see [KI]). It was generally assumed for a long time that zeros of the grand

partition function lie on a smooth curve. But in 1983, it was realized that the

picture of the distribution of this kind of zeros is not so simple. Derrida, De Seze

and Itzykso ([DDI]) found fractal patterns in so-called hierarchical lattices. It has

been shown for many examples that these singularities are located on the Julia set

associated with a renormalization transformation (see [DDI], [MO2], [PR]). Some

interesting relationships between critical exponents, critical amplitudes and the

shape of a Julia set have been found ([DIL]). In [BL], Bleher and Lyubich studied

Julia sets and complex singularities in diamond-like hierarchical Ising models. For

a general model, they reformulated the following problem: How are singularities of

the free energy continued to the complex space and what is their global structure

in the complex space?

In this article, we deal with a λ-state Potts model on a generalized diamond

hierarchical lattice which is a natural generalization of a diamond-like hierarchical

Ising models studied in many papers in the past thirty years (see [BL], [DDI],

[DIL], [PR], [QI5], [QL], [QYG], [YA]). A λ-state Potts model (for integer or

non-integer values of λ) plays an important role in the general theory of phase

transitions and critical phenomena ([GU], [HU], [LI], [OS]). In this article, it is

proved that the limit distribution of complex singularities of the free energy of a

generalized diamond hierarchical Potts model is exactly the Julia set of a renor-

malization transformation with three parameters (Theorem 1.1). The main subject

of this article is the structure of this family of Julia sets. In view of the problem

concerning the distribution of complex singularities proposed in [YL] and [BL], we

give a complete description about the connectivity and the local connectivity of

these Julia sets (Theorem 3.1-3.3, Theorem 4.1). One of significant results is that

the Julia set of the renormalization transformation for some parameters contains a

small Feigenbaum Julia set which intersects with the positive real axis in a closed

interval (Theorem 2.2). This is an interesting phenomenon which has never been

found before. Since the positive real axis corresponds to the real world, it may lead

to new problems in the research of statistical physics. In order to deal with the

free energy on the Riemann sphere, we study the regularity of boundaries of all

components of the Fatou set of the renormalization transformation (Theorem 4.2

and Theorem 4.3). These results will help in the study of the boundary behavior

of the free energy. Finally, an explicit value of the second order critical exponent of

the free energy for almost all points on the boundary of the immediately attractive

basin of infinity is given (Theorem 5.4).

In this article, we shall use Umnλ to denote the above renormalization trans-

formation, where m, n ∈ N and λ ∈ R are three parameters. In Chapter 1, we

introduce basic notations and fundamental results in complex dynamical systems.

We also give a definition of a generalized diamond hierarchical Potts model. By a

classical theorem in the theory of complex dynamical systems we can deduce that

the set of complex singularities of a generalized diamond hierarchical Potts model

is the Julia set of the renormalization transformation Umnλ (Theorem1.1).

Chapter 2 is devoted to study the dynamical complexity of renormalization

transformations Umnλ with variant parameters m, n and λ. Firstly, we give a

marvellous factorization of Umnλ. It is very helpful to us for dealing with the