Intersection of orbits of loxodromic automorphisms of affine surfaces

TL;DR

This paper proves that two loxodromic automorphisms of an affine surface sharing an infinite orbit must have a common iterate, and establishes a dynamical Mordell-Lang type result for such surfaces.

Contribution

It demonstrates a new rigidity property of loxodromic automorphisms on affine surfaces and extends dynamical Mordell-Lang results to this setting.

Findings

Two automorphisms with infinitely intersecting orbits share a common iterate

Established a dynamical Mordell-Lang type theorem for affine surface products

Connected automorphism dynamics with arithmetic dynamics techniques

Abstract

We show the following result: If $X_0$ is an affine surface over a field $K$ and $f, g$ are two loxodromic automorphisms with an orbit meeting infinitely many times, then $f$ and $g$ must share a common iterate. The proof uses the preliminary work of the author in [Abb23] on the dynamics of endomorphisms of affine surfaces and arguments from arithmetic dynamics. We then show a dynamical Mordell-Lang type result for surfaces in $X_0 \times X_0$.