Rational orbits under correspondences

Consider an algebraic function like $F(x)=\sqrt{x^3-1}$. If $p\in \mathbb{Q}$ is a rational number, how many iterates of $p$ under $F$ can also be rational? The dynamics of algebraic functions may be formalized in the language of correspondences on curves and their iterates. In this paper we show that if $F$ is a correspondence from ${\mathbb{P}}^1$ to itself defined over a finitely generated field $K$ of characteristic 0 satisfying several minor constraints, then either for each $n\ge 12$ there are only finitely many $p\in \mathbb{Q}$ for which $F^n(p)$ contains a $K$-rational point or $F$ belongs to an explicit list of known exceptional correspondences.