The Geometric Dynamical Northcott Property For Regular Polynomial Automorphisms of the Affine Plane

TL;DR

This paper proves the finiteness of periodic points for regular polynomial automorphisms of the affine plane over function fields, linking canonical heights with bifurcation currents and stability of points.

Contribution

It establishes the Geometric Dynamical Northcott Property for these automorphisms, improving previous results by connecting heights, bifurcation currents, and stability.

Findings

Finiteness of periodic points over function fields.

Canonical height equals bifurcation current mass.

Stability of points characterized by zero canonical height.

Abstract

We establish the finiteness of periodic points, that we called Geometric Dynamical Northcott Property, for regular polynomials automorphisms of the affine plane over a function field $\mathbf{K}$ of characteristic zero, improving results of Ingram. For that, we show that when $\mathbf{K}$ is the field of rational functions of a smooth complex projective curve, the canonical height of a subvariety is the mass of an appropriate bifurcation current and that a marked point is stable if and only if its canonical height is zero. We then establish the Geometric Dynamical Northcott Property using a similarity argument.