6 1. COMPLEX DYNAMICS AND POTTS MODELS

Theorem A1. J(R) = ∅ and J(R) is a perfect set.

Theorem A2. If J(R) has interior points, then J(R) =

¯.

C

For any point z0 ∈

¯,

C the backward orbit of z0 is defined to be the set

O−(z0)

= {z :

Rj(z)

= z0 for some j ≥ 0}.

If O−(z0) is a finite set, then z0 is called an exceptional point of R. Denote the set

of exceptional points of R by ER. We have

Theorem A3. The set ER contains at most two points. Every point in ER is

a superattractive periodic point.

Theorem A4. For arbitrary a0 ∈

¯

C \ ER, the Julia set J(R) belongs to the

limiting set of the backward orbit of a0. Furthermore, if a0 ∈ J(R), then the limiting

set of the backward orbit of a0 is indeed the Julia set J(R).

Theorem A5. The number of components of F (R) may only be 0, 1, 2 or ∞.

If D is a completely invariant component of F (R), then D contains at least d − 1

critical points, all components of F (R) \ D are simply connected and J(R) = ∂D.

Furthermore, if F (R) has two completely invariant components D and D0, then

F (R) = D ∪ D0, D and D0 are two Jordan domains, and J(R) is a Jordan curve.

Theorem A6. The Julia set J(R) is the closure of all repulsive periodic points

of R.

Theorem A7. Let V be an open set and K be a compact set. If

V ∩ J(R) = ∅ and K ∩ ER = ∅,

then there exists a constant N 0 such that Rn(V ) ⊃ K for n ≥ N.

Theorem A8 (Sullivan’s non-wandering domain theorem, see [SU]). Each

component of F (R) is eventually periodic under the iteration of R. If D is a forward

invariant component of F (R), then there are just five possibilities:

(i) D is the immediate basin of an attractive fixed point, i.e., D is a component

of F (R) containing an attractive fixed point;

(ii) D is the immediate basin of a superattractive fixed point, i.e., D is a com-

ponent containing a superattractive fixed point;

(iii) D is the immediate basin of a parabolic fixed point, i.e., there is a parabolic

fixed point z0 ∈ ∂D such that Rn(z) → z0 as n → ∞ for z ∈ D;

(iv) D is a Siegel disc, i.e., R : D → D is analytically conjugate to a Euclidean

rotation of the unit disc onto itself;

(v) D is a Herman ring, i.e., R : D → D is analytically conjugate to a Euclidean

rotation of some annulus onto itself.

Furthermore, if D is one of the types (i), (ii) and (iii), then D contains at least

one critical point of R; if D is one of the types (iv) and (v), then every boundary

point of D belongs to the closure of the forward orbit of some critical point of R.

For further results in complex dynamics, see [CG] or [MI].

1.2. Julia sets related to Potts models

In this section, we introduce λ-state Potts models on generalized diamond hi-

erarchical lattices. It can be shown that limiting sets of zeros of grant partition