A dynamical characterisation of smooth cubic affine surfaces of Markov type

TL;DR

This paper classifies certain smooth affine surfaces with complex automorphism groups, showing they are either algebraic tori or specific cubic surfaces of Markov type, and studies their endomorphism properties.

Contribution

It provides a classification of smooth cubic affine surfaces of Markov type using valuative techniques and analyzes their endomorphism structure.

Findings

Surfaces are either algebraic tori or of Markov type.

Markov type surfaces do not admit non-automorphism dominant endomorphisms.

Classification based on automorphism group properties.

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.