Complex dynamics and Potts models
We introduce in this chapter some standard notation and some fundamental
results in the theory of iterations of rational maps. These notations and results will
be constantly used throughout the whole article. We briefly introduce the concept
of a partition function and the concept of a renormalization transformation in sta-
tistical mechanics. After defining a generalized diamond hierarchical Potts model,
we establish a relationship between the unstable set of the complex dynamical sys-
tem of the renormalization transformation and the set of complex singularities of
the free energy.
1.1. Iterations of a rational map
Suppose R is a rational map of degree larger than one from the Riemann
C to itself. We denote the j-th iterate of R by
For any point z0
the sequence
is called the orbit of z0, denote it by
The Fatou
set is defined by
F (R) = {z
C |
is normal at z},
and the Julia set J(R) of R is the complement of the Fatou set F (R), i.e.,
J(R) =
C \ F (R).
Obviously, J(R) is a closed set and is completely invariant. F (R) is an open
set which contains countably many domains. Furthermore, if F (R) = ∅, then
∂F (R) = J(R) and each component of F (R) is called a Fatou component.
Let z0
C such that
= z0 and
(z0) = z0 for j = 1, 2,...,p 1. Then
z0 is called a periodic point of period p, and the set
is called a periodic orbit or cycle (of period p). If p = 1, then z0 is called a fixed
point. In order to characterize the stability of a periodic point z0 of period p, one
computes the derivative λ =
(z0) which is called the eigenvalue of z0. z0 is
called superattractive, attractive, repulsive or indifferent according to
λ = 0, 0 |λ| 1, |λ| 1 or |λ| = 1
respectively. Furthermore, if λ =
and θ [0, 1) is a rational number, then z0 is
called a parabolic periodic point (or rationally indifferent periodic point). We know
that all attractive and superattractive periodic points of a rational map R belong
to the Fatou set F (R), while all repulsive and parabolic periodic points belong to
the Julia set J(R).
The following are some fundamental properties in complex dynamics, which we
just list without proofs. The reader who is interested in them can refer to [CG] or
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