Tripling function and its periodic points

Question

Given tripling function

$$F(x)= \begin{cases}3x & 0<x<1/3 \\ 3x-1 & 1/3<x<2/3 \\ 3x-2 & 2/3<x<1 \end{cases}$$

Suppose $x=\frac{p}{q}$ is rational number. Show that $x$ is either periodic or eventually periodic for $F$.

Can anyone help me?

Answer 1

Edit: In light of the closure of the question, I should have first asked you what efforts have you made?

If $x$ is periodic there is nothing to prove so assume that $x$ is not periodic.

No matter what 'subdomain' $x=\frac{p}{q}$ falls into, $F(x)$ is still in the set

$$\{t\cdot \frac{1}{q}\,:\,t=0,1,2,\dots q\}.$$

Prove this.

Apply the tripling function $q+1$ times. Now the set of $q+2$ iterates (including $x$) lies in a set of $q+1$ elements. What can you conclude?