8 1. COMPLEX DYNAMICS AND POTTS MODELS

First, we point out that the map Umnλ is actually the renormalization transfor-

mation of the generalized diamond hierarchical Potts model. It is well known that

the idea of renormalization formulated by Wilson (see [WI1], [WI2]) in statistical

mechanics can be understood as a successive thinning out of the degree of freedom

in the partition function. The j0-particle problem is transformed into a j1-particle

problem with j1 j0. The precise formulation of the renormalization idea is to

find a transformation

R : T → R(T )

such that the j0-particle partition function can be expressed by the simpler j1-

particle partition function. The critical point is a repulsive fixed point of the

renormalization transformation. This is the basic reason on which Wilson derived

the critical power laws (see [WI1], [WI2]). However, in general it is diﬃcult to

find the renormalization transformation. The advantage of hierarchical lattices is

that one can find the renormalization transformation of the model exactly.

For the generalized diamond hierarchical Potts model, we can deduce that the

transformation Umnλ is indeed the recursion relation between two levels. Consider-

ing the relation between the Julia set of Umnλ and the set of zeros of the partition

function, we have:

Theorem 1.1. Let CU be the limiting set of zeros of the partition function

of a λ-state Potts model on the generalized diamond hierarchical lattice. If −λ +

1 is not a periodic point of Umnλ, then J(Umnλ) = CU ; otherwise, J(Umnλ) =

CU O+(−λ + 1).

Proof. First, in order to derive the recursion relation for the partition func-

tion, we look at the first two levels of constructions Γ1 and Γ2. By using the

Migdal-Kadanoff renormalization procedure (see [KA]), we easily obtain that

Z2(z) =

(z + λ −

1)n

− (z −

1)n

λ

m

· Z1(ξ)

here

ξ = Umnλ(z) =

(z + λ −

1)n

+ (λ − 1)(z −

1)n

(z + λ − 1)n − (z − 1)n

m

.

The general recursion relation between partition functions at the (j − 1)-th level

and the j-th level for j ≥ 2 is

(1.3) Zj(z) =

(z + λ −

1)n

− (z −

1)n

λ

mj−1nj−2

· Zj−1(ξ).

Since there are

(mn)j−1

bonds in Γj, by (1.1) we can easily see that Zj(z) is a

polynomial of degree

(mn)j−1

and it can be expressed by

Zj(z) = λ

(mn)j−1

k=1

(z − zk),

here zk is a zero of Zj (z). By (1.2) and (1.3), we have

(mn)j−1

k=1

(z − zk)

=

(mn)j−2

k=1

{[(z + λ −

1)n

+ (λ − 1)(z −

1)n]

− zk[(z + λ −

1)n

− (z −

1)n]},