The dynamical Manin–Mumford problem for plane polynomial automorphisms

Abstract

Let $f$ be a polynomial automorphism of the affine plane. In this paper we consider the possibility for it to possess infinitely many periodic points on an algebraic curve $C$. We conjecture that this happens if and only if $f$ admits a time-reversal symmetry; in particular the Jacobian Jac $(f)$ must be a root of unity.

As a step towards this conjecture, we prove that the Jacobian and all its Galois conjugates lie on the unit circle in the complex plane. Under mild additional assumptions we are able to conclude that indeed Jac $(f)$ is a root of unity.

We use these results to show in various cases that any two automorphisms sharing an infinite set of periodic points must have a common iterate, in the spirit of recent results by Baker–DeMarco and Yuan–Zhang.