The dynamical Manin–Mumford problem for plane polynomial automorphisms

Romain Dujardin,Charles Favre

doi:10.4171/jems/743

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 hap- pens 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 its Jacobian, together with 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 automor- phisms sharing an infinite set of periodic points must have a common it- erate, in the spirit of recent results by Baker-DeMarco and Yuan-Zhang.

Highlights

In this paper we discuss the following problem in the case of polynomial automorphisms of the affine plane.Dynamical Manin-Mumford Problem
In case (X, f ) is the dynamical system induced on an abelian variety by the multiplication by an integer ≥ 2, it is a deep theorem originally due to M
Our goal is to explore this problem when f is a polynomial automorphism of the affine plane A2, defined over a field of characteristic zero

Summary

Introduction

In this paper we discuss the following problem in the case of polynomial automorphisms of the affine plane. Even though this problem seems very delicate, we are able to address some cases of the dynamical Manin-Mumford problem for special product maps of Henon type, as the following theorem shows. Theorem A implies that if | Jac(f )| = 1 it is enough to assume that f and g share an infinite set of periodic points to conclude that f n = gm. Otherwise we use the equality of equilibrium measures at all places to infer that f and g, as well as any automorphism belonging to the group generated by f and g, have the same sets of periodic points.

Polynomial automorphisms over a metrized field

Applying the equidistribution theorem

The DMM statement in the complex dissipative case

The DMM statement under a transversality assumption

Automorphisms sharing periodic points

Reversible polynomial automorphisms