Rigidity of periodic points for loxodromic automorphisms of affine surfaces

TL;DR

This paper establishes a criterion for when two automorphisms of an affine surface with high dynamical degree share a dense set of periodic points, introducing canonical heights and applying arithmetic equidistribution techniques.

Contribution

It provides a characterization of shared periodic points for automorphisms with dynamical degree greater than one, using canonical heights and equidistribution methods over various fields.

Findings

Automorphisms share a Zariski dense set of periodic points iff they have identical periodic points.

Construction of canonical heights for these automorphisms.

Application of arithmetic equidistribution and Moriwaki heights across different base fields.

Abstract

We show that two automorphisms of an affine surface with dynamical degree strictly larger than 1 share a Zariski dense set of periodic points if and only if they have the same periodic points. We construct canonical heights for these automorphisms and use arithmetic equidistribution for adelic line bundles over quasiprojective varieties following the work of Yuan and Zhang. When the base field is not a number field or the function field of a curve we use the theory of Moriwaki heights to prove the result.