CHAPTER 1

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

sphere

¯

C to itself. We denote the j-th iterate of R by

Rj.

For any point z0 ∈

¯,

C

the sequence

{Rj(z0)}j=0 ∞

is called the orbit of z0, denote it by

O+(z0).

The Fatou

set is defined by

F (R) = {z ∈

¯

C |

{Rj}

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

Rp(z0)

= z0 and

Rj

(z0) = z0 for j = 1, 2,...,p − 1. Then

z0 is called a periodic point of period p, and the set

O+(z0)

=

{z0,R(z0),...,Rp−1(z0)}

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 =

(Rp)

(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 =

ei2πθ

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

[MI].

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