Rational orbits under correspondences

TL;DR

This paper investigates the rational iterates of algebraic functions viewed as correspondences on curves, establishing finiteness results under certain conditions and classifying exceptions.

Contribution

It proves finiteness of rational iterates for a broad class of correspondences and explicitly classifies known exceptional cases.

Findings

Finiteness of rational points for iterates n ≥ 12 under certain correspondences.

Classification of known exceptional correspondences.

Conditions under which the finiteness result holds.

Abstract

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 $n \geq 12$ 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.