1.2. JULIA SETS RELATED TO POTTS MODELS 7

functions are Julia sets of a family of rational maps, which are renormalization

transformations with respect to these physical models.

The hierarchical sequence of a generalized diamond lattice depends on two

natural parameters m ≥ 2 and n ≥ 2. The lattice Γ1 is just two ”outer” sites

related by a bond. In order to obtain Γ2, we connect two outer sites by m branches

of n bonds. Furthermore, in order to obtain Γj+1, we replace each bond in Γj by

the lattice Γ2 (j = 1, 2, · · · ) (see Fig 1.1 for m = 2 and n = 3).

Fig 1.1. Γ1, Γ2 and Γ3 for m = 2 and n = 3.

The above generalized diamond lattice is obviously a generalization of the stan-

dard diamond lattice (m = n = 2) and the diamond-like lattice (m ∈ N and n = 2).

The Potts model was initially defined for an integer λ as a generalization of the

Ising model (λ = 2) to more than two components (see [HU]). Later on, it was

shown that the Potts model for non-integer values λ may describe properties of a

number of physical systems such as dilute spin glasses, gelation and vulcanization

of branched polymers (0 λ 1). Also it was shown that bond and site percola-

tion problems could be formulated in terms of Potts models with pair and multisite

interactions in the λ = 1 limit (see [GA], [GU], [LI], [MO1], [MO2]).

At each site of the generalized diamond lattice we put a Potts spin which

can have λ different states, we then get a λ-state Potts model on the generalized

diamond hierarchical lattice. The Hamiltonian of the λ-state Potts model is

H = −J

ij

δ(σi,σj), σi = 1, 2, · · · , λ

where δ is the Kronecker delta, and the sum is over nearest neighbors, and J is the

exchange interaction constant. The partition function is then given by

(1.1) Z =

{σi}

exp [K

ij

δ(σi,σj)],

where K = J

kT

, here k is the Boltzmann constant and T is the temperature. This

describes ferromagnetic interaction for J 0, and antiferromagnetic interaction for

J 0 (see [HU]).

For convenience, we use the variable z = eK . In the following, we shall show

that the limiting set of the zeros of the partition function is the Julia set J(Umnλ)

of the rational map

(1.2) w = Umnλ(z) =

(z + λ −

1)n

+ (λ − 1)(z −

1)n

(z + λ − 1)n − (z − 1)n

m

.