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