[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
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