SILK-ROAD 2003 Problem 3

Let $0<a<b<1$ be reals numbers and $$g(x)=\left\{\begin{array}{cc}x+1-a,&\text{ if } 0<x<bb-a, & \text{ if } x=a x-a, & \text{ if } a<x<b1-a ,&\text{ if } x=b x-a ,&\text{ if } b<x<1 \end{array}\right.$$ Give that there exist $n+1$ reals numbers $0<x_0<x_1<...<x_n<1$ , for which $g^{[n]}(x_i)=x_i \ (0 \leq i \leq n)$ . Prove that there exists a positive integer $N$ , such that $g^{[N]}(x)=x$ for all $0<x<1$ .

( $g^{[n]}(x)= \underbrace{g(g(....(g(x))....))}_{\text{n times}}$ )

Official solution here