Rigidity of periodic points for loxodromic automorphisms of affine surfaces
Marc Abboud
TL;DR
The paper establishes a rigidity principle for loxodromic automorphisms of normal affine surfaces: two such maps have a Zariski dense intersection of periodic points if and only if they share the same periodic points, with the main tool being canonical heights built from invariant adelic divisors. It develops a comprehensive arithmetic-dynamics framework, including Green functions, invariant adelic divisors, and Moriwaki heights to handle bases that are not number fields, and applies arithmetic equidistribution to relate height-zero sets to dynamical invariants. A complete analysis is given of the algebraic torus as a boundary case, and the work extends to geometric settings, yielding strong rigidity results for Henon maps and Markov-type surfaces via equilibrium measures. The results contribute a robust affine-surface analogue of projective rigidity theorems, with potential implications for unlikely intersections in higher-dimensional dynamics and algebraic dynamics on nonprojective varieties.
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.