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On the dynamics of endomorphisms of affine surfaces

Marc Abboud

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.

On the dynamics of endomorphisms of affine surfaces

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.
Paper Structure (110 sections, 194 theorems, 423 equations, 9 figures)

This paper contains 110 sections, 194 theorems, 423 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Dynamical degrees
  3. Dynamical degrees on projective surfaces
  4. Dynamical degrees of endomorphisms of affine surfaces and Perron numbers
  5. The dynamical spectrum of the algebraic torus
  6. Existence of an eigenvaluation
  7. Discussion on the assumptions of Theorem \ref{['BigThmExistenceValuationPropreEng']}
  8. Statement of the theorem in the case of automorphisms
  9. Valuations and Algebraic geometry
  10. Results from Algebraic Geometry
  11. Bertini
  12. Local power series and local coordinates
  13. Boundary
  14. Surfaces
  15. Rigid contracting germs in dimension 2 and local normal forms
  16. ...and 95 more sections

Key Result

Theorem 1.1

favreEigenvaluations2007 Let $f : \mathbf{C}^2 \rightarrow \mathbf{C}^2$ be a dominant polynomial transformation, then $\lambda_1 (f)$ is a Perron number of degree $\leq 2$.

Figures (9)

  • Figure 1: The homeomorphism between $\cV_\mathfrak m$ and $\cV_z$
  • Figure 2: Algorithm for computing the generic multiplicity
  • Figure 3: The endomorphism $f$ on $X_0$
  • Figure 4: The embedding $\pi_\bullet$
  • Figure 5: Configuration which is not possible
  • ...and 4 more figures

Theorems & Definitions (365)

  • Theorem 1.1
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Theorem 2.1: Bertini's Theorem, hartshorneAlgebraicGeometry1977
  • Theorem 2.2: hartshorneAlgebraicGeometry1977, Theorem 5.5A
  • Proposition 2.3: goodmanAffineOpenSubsets1969, Proposition 1 and 2
  • ...and 355 more