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Holomorphically conjugate polynomial automorphisms of C^2 are polynomially conjugate

Serge Cantat, Romain Dujardin

Abstract

We confirm a conjecture of Friedland and Milnor: if two polynomial automorphisms f and g in Aut(C^2) with dynamical degree >1 are conjugate by some holomorphic diffeomorphism φof C^2, then φis a polynomial automorphism; thus, f and g are conjugate inside Aut(C^2). We also discuss a number of variations on this result.

Holomorphically conjugate polynomial automorphisms of C^2 are polynomially conjugate

Abstract

We confirm a conjecture of Friedland and Milnor: if two polynomial automorphisms f and g in Aut(C^2) with dynamical degree >1 are conjugate by some holomorphic diffeomorphism φof C^2, then φis a polynomial automorphism; thus, f and g are conjugate inside Aut(C^2). We also discuss a number of variations on this result.
Paper Structure (7 sections, 3 theorems, 8 equations)

This paper contains 7 sections, 3 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Conjugacy classes in $\mathsf{Aut}(\mathbb{A}^2_{\mathbf{K}})$
  3. Conjugation under biholomorphisms and rigidity
  4. Proof of Theorem A
  5. Preliminaries on Hénon maps (see friedland-milnor and Sibony:Panorama)
  6. The proof
  7. Discussion and complements

Key Result

Lemma 1

There exists a positive real number $c$ such that $\varphi^\varstar T_g^+ = c T_f^+$ and $G_g^+\circ \varphi = c G_f^+$.

Theorems & Definitions (8)

  • Lemma
  • proof
  • Lemma
  • proof
  • Example
  • proof
  • Lemma
  • proof : Sketch of proof