CMO 1989 Problem 3

Let $S$ be the unit circle in the complex plane (i.e. the set of all complex numbers with their moduli equal to $1$ ). We define function $f:S\rightarrow S$ as follow: $\forall z\in S$ , $f^{(1)}(z)=f(z), f^{(2)}(z)=f(f(z)), \dots,$ $f^{(k)}(z)=f(f^{(k-1)}(z)) (k>1,k\in \mathbb{N}), \dots$ We call $c$ an $n$ -period-point of $f$ if $c$ ( $c\in S$ ) and $n$ ( $n\in\mathbb{N}$ ) satisfy: $f^{(1)}(c) \not=c, f^{(2)}(c) \not=c, f^{(3)}(c) \not=c, \dots, f^{(n-1)}(c) \not=c, f^{(n)}(c)=c$ . Suppose that $f(z)=z^m$ ( $z\in S; m>1, m\in \mathbb{N}$ ), find the number of $1989$ -period-point of $f$ .