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
Previous Page Next Page