Introduction
The theory of complex dynamical systems is the study of a dynamical system
generated by an non-invertible analytic map R : S S of a Riemann surface
S. An important example is a rational map of the Riemann sphere C. In this
case the dynamical system is the semigroup of iterations Rj (j = 1, 2,...). The
basic problem is to understand the phase portrait of such a system, that is, the
typical behavior of orbits
{Rj(z)}j=0,
as well as the character of change of the
phase portrait under the deformation of R.
It is well known that the theory of complex dynamical systems was first studied
at the beginning of the last century by G. Julia ([JU]) and P. Fatou ([FA1], [FA2]).
During 1919 to 1921, by applying the theory of normal families to the theory of
iterations of analytic maps, they established the foundation of the theory of complex
dynamical systems which is called the Fatou-Julia theory now. However, this theory
passed through a fifty-year epoch of stagnation before 1980, when it entered a period
of great development by introduceing modern techniques into the study. It focuses
in itself ideas and methods of very diverse areas of mathematics. This kind of
dynamical systems provides an understanding of the nature of chaos, the fractal
property, and structural stability. Thus it has became one of the main sources for
the formulation of problems in the nonlinear theory.
For a complex dynamical system, the Julia set is an unstable set, while the
Fatou set is a stable set. It is well known that a typical Julia set is fractal, the
dynamical system on the Julia set is chaotic. In 1983, Derrida, De Seze, and Itzykso
([DDI]) found a connection between the phase transition in statistical mechanics
and Julia sets in complex dynamical systems. In 1952, Yang and Lee proved the
celebrated Yang-Lee theorem ([LY], [YL]) in statistical mechanics. The theorem
deals with the analytic continuation of the free energy on the complex plane, here
the free energy means the logarithm of the partition function. They studied the
distribution of zeros of the partition function which is considered as a function of a
complex magnetic field (Yang-Lee zeros). They proved the famous circle theorem
which states that zeros of the partition function of an Ising ferromagnet lie on the
unit circle in an externally applied complex magnetic field plane. Hence complex
singularities of the free energy lie on the unit circle as well. After this pioneer work,
Fisher ([FI]) in 1964 initiated the study of zeros of the partition function in the
complex temperature plane (Fisher zeros). These methods were then extended to
other types of interactions and found a wide range of applications (see [BO], [GA],
[GU], [KI], [LI], [MO1], [MO2]).
An important problem stated in [YL] is to study the limit distribution of zeros
of the grand partition function. The reason is that the free energy can be expressed
as a logarithmic potential over this distribution. Since 1952, numerous articles have
dealt with various properties of complex singularities of ferromagnetic models (see
1
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