The dynamical Manin-Mumford problem for plane polynomial automorphisms

TL;DR

This paper investigates conditions under which polynomial automorphisms of the affine plane have infinitely many periodic points on a curve, linking this to symmetries and properties of the Jacobian, and extends dynamical results to these automorphisms.

Contribution

It establishes a connection between infinite periodic points on a curve and time-reversal symmetry in polynomial automorphisms, advancing the understanding of the dynamical Manin-Mumford problem.

Findings

Jacobian and its Galois conjugates lie on the unit circle

Under mild assumptions, Jacobian is a root of unity

Automorphisms with infinite shared periodic points have a common iterate

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 $\mathrm{Jac}(f)$ must be a root of unity. As a step towards this conjecture, we prove that the Jacobian of $f$ 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 $\mathrm{Jac}(f)$ 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.