MathDB

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

n

-period-point of

f

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

Problem Statement