A dynamical characterisation of smooth cubic affine surfaces of Markov type

2025/12/11 by Marc Abboud, Abboud, Marc
Mathematics · #13A18 #37F10 #37F80 #37P99 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2512.10455

openalex publication_date 2025/12/11 · openalex created_date 2025/12/13 · openalex updated_date 2026/07/28

Abstract

Using valuative techniques, we show that a smooth affine surface with a non-elementary automorphism group and completable by a cycle of rational curves is either the algebraic torus or a smooth cubic affine surface of Markov type. Furthermore we show that smooth cubic affine surfaces of Markov type do no admit dominant endomorphisms which are not automorphisms.

Citations

• Intersection of orbits of loxodromic automorphisms of affine surfaces
• On the dynamics of endomorphisms of affine surfaces
• A smooth complex rational affine surface with uncountably many real forms
• Affine surfaces with a huge group of automorphisms
• Holomorphic dynamics, Painlevé VI equation and Character Varieties
• Degree growth of meromorphic surface maps

Related

• On the dynamics of endomorphisms of affine surfaces
• Characterization of affine surfaces with a torus action by their automorphism groups
• Markoff Triples and Strong Approximation
• Automorphisms of cubic surfaces in positive characteristic
• Affine cubic surfaces and relative SL(2)-character varieties of compact\n surfaces