1.2. JULIA SETS RELATED TO POTTS MODELS 9

here zk is a zero of Zj−1(z). Therefore, the set of zeros of Zj is just the preimage

of the set of zeros of Zj−1 under the map Umnλ. By (1.1), we know

Z1(z) = λ(z + λ − 1)

has only one zero −λ + 1. So we can obtain

(mn)j−1

zeros of Zj by working out

the preimage of the unique zero −λ + 1 of Z1 under the map

Umnλ ◦ Umnλ ◦ · · · ◦ Umnλ (j − 1 times).

If −λ + 1 ∈ J(Umnλ), by Theorem A4, CU = J(Umnλ); If −λ + 1 ∈ F (Umnλ),

{Umnλ(−λ

−j

+ 1)}j=1

∞

is obviously an infinite set. It is easy to verify that −λ + 1 is a

critical point of Umnλ. By Theorem A8, −λ + 1 can not belong to a Siegel disc or

an Hermann ring, so

{Umnλ(−λ

−j

+

1)}j=1∞

has no limiting point on F (Umnλ) provided that −λ +1 is not a periodic point. We

thus complete the proof of Theorem 1.1.