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arXiv · 2409.07826

Intersection of orbits of loxodromic automorphisms of affine surfaces

Marc Abboud
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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 $X_0 \times X_0$.

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Marc Abboud. 2024-09-12. Intersection of orbits of loxodromic automorphisms of affine surfaces. https://arxiv.org/abs/2409.07826

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