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
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),
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
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.