For algebraic correspondences F on P1 over finitely generated K of char 0 (with minor constraints), either there are only finitely many rational p with F^n(p) containing a K-rational point for all n >= 12, or F is one of a known list of exceptional correspondences.
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Rational orbits under correspondences
For algebraic correspondences F on P1 over finitely generated K of char 0 (with minor constraints), either there are only finitely many rational p with F^n(p) containing a K-rational point for all n >= 12, or F is one of a known list of exceptional correspondences.