Consider a function $f\colon\mathbb{Q}_{>0}\longrightarrow\mathbb{Q}_{>0}$ such that
$f(x)=\frac{3m-1}{2n+1}$ where $x=\frac{m}{n}$ and $\frac{m}{n}$ is reduced form. (i.e., $\gcd(n,m)=1$ and $n,m\in\mathbb{Z}_{>0}$) Questions are,
(1) For which positive rational number $x$, $f(x)<x$?
(2) Let $x_0\in\mathbb{Q}_{>0}$. Then, construct a sequence $x_1=f(x_0),x_2=f(x_1),\dots,x_k=f(x_{k-1}),\dots$.
For which $x_0$, there exists a positive integer $k$ such that $x_k=x_0$?
(In other words, for which $x_0$ the sequence has periodicity?)
-Progress so far: (1) is easy. For $x=m$ or $\frac{1}{n}$ for some $n,m\in\mathbb{Z}_{>0}$ we have $f(x)<x$.
But for the problem (2), I have no clue. I only know some special cases. If $x_0=2$ or $\frac{1}{3}$ the sequence has period $2$. But I don't know how to derive general solutions.