On the dynamics of endomorphisms of affine surfaces

2023/11/30 by Abboud, Marc · 2 citations
#Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2311.18381

Abstract

In [FJ07], Favre and Jonsson developed tools from valuative theory to study the dynamics of a dominant endomorphism of the complex affine plane. We extend this theory to the case of any affine surface, over any field. We give a new method to construct an eigenvaluation of an endomorphism. We generalize the result of Favre and Jonsson and show that the first dynamical degree of a dominant endomorphism of anormal affine surface is an algebraic integer of degree at most 2. Plus, we obtain a new result of rigidity. The set of first dynamical degrees of loxodromic automorphisms of a given affine surface must be contained in the set of integers or in the set of algebraic numbers of degree 2.

Cited by

• A dynamical characterisation of smooth cubic affine surfaces of Markov type
• Rational self-maps of projective surfaces with a regular iterate

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