I have a rational function, $f(x)$, and I would like to prove that for all positive integers, $i,j$, and for all rationals, $w=q/p$, such that $w \notin \{ -1, 0, 1 \}$, we have $f^i(w) = f^j(w)$ only if $i=j$ or, in other words, that the sequence $w_i=f^i(w)$ is acyclic. The superscripts here refer to composition, not multiplication.
We can reduce it to two polynomials, $\frac{f_1}{ f_2}$ in the integer variables $p, q$.
We can then show that:
1) For all $p,q$, we have that $gcd(f_1 (p, q), f_2(p, q))$ is some power of $2$.
2) $pq|f_1(p,q) $ and $gcd(\frac {f_1(p,q)}{pq}, pq) \in \{1, 2, 4 \}$.
3) Neither $\frac {f_1(p,q)}{pq}$ nor $f_2(p,q)$ is $1$ and that at most one of them is a power of $2$.
Does this suffice to prove that the sequence $w_i=f^i(w)$ is acyclic? Or am I missing something?