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Multiplier rigidity for complex Hénon maps

Serge Cantat, Romain Dujardin

Abstract

We investigate the multiplier rigidity problem for polynomial automorphisms of $\mathbf{C}^2$. A first result states that a complex Hénon map of given degree is determined up to finitely many choices by its multiplier spectrum, or more generally by the unstable multipliers of its saddle periodic points. This is the counterpart in this setting of a classical result of McMullen for one-dimensional rational maps. For compositions of Hénon maps, the same rigidity holds provided the multi-degree and the multi-Jacobian are fixed. As in McMullen's theorem, this follows from the nonexistence of stable algebraic families in the corresponding parameter space. This in turn relies on precise asymptotic bounds for the Lyapunov exponents of the maximal entropy measure along diverging families.

Multiplier rigidity for complex Hénon maps

Abstract

We investigate the multiplier rigidity problem for polynomial automorphisms of . A first result states that a complex Hénon map of given degree is determined up to finitely many choices by its multiplier spectrum, or more generally by the unstable multipliers of its saddle periodic points. This is the counterpart in this setting of a classical result of McMullen for one-dimensional rational maps. For compositions of Hénon maps, the same rigidity holds provided the multi-degree and the multi-Jacobian are fixed. As in McMullen's theorem, this follows from the nonexistence of stable algebraic families in the corresponding parameter space. This in turn relies on precise asymptotic bounds for the Lyapunov exponents of the maximal entropy measure along diverging families.
Paper Structure (41 sections, 39 theorems, 128 equations, 2 figures)

This paper contains 41 sections, 39 theorems, 128 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview of the problem
  3. Classification of polynomial automorphisms
  4. Multiplier rigidity
  5. Stable algebraic families
  6. Divergence of Lyapunov exponents
  7. Logical structure
  8. Stability theory of loxodromic automorphisms
  9. Dynamics of loxodromic automorphisms
  10. $J^\varstar$-stability
  11. A worked out example
  12. Existence of bifurcations
  13. The fixed point $\alpha_a$ and bifurcations of type (V)
  14. Bifurcations in the unit disk
  15. Periodic points and multiplier spectra
  16. ...and 26 more sections

Key Result

Theorem 1

A complex Hénon map $f(x,y)=(ay+p(x),x)$ is determined up to finitely many choices by its trace spectrum (resp. its unstable multiplier spectrum).

Figures (2)

  • Figure 1: Slice of the filled Julia set of $f_{-0.669+0.73 i}$ by the line $\left\{y=0\right\}$. The basin of $(0,0)$ is colored black, and the basin of the period-3 cycle is in red. Detail of the picture on the right.
  • Figure 2: Proof of Lemma \ref{['lem:folding']}. The boundaries of the bidisks $\mathbb{B}'_2$ and $\mathbb{B}_3$ are dotted and the solenoidal component of $f(\mathbb{B}_1)\cap \mathbb{B}_2$ is shaded.

Theorems & Definitions (94)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 1.1
  • Theorem 5
  • Theorem 2.1
  • Proposition 2.2
  • Remark 2.3
  • proof : Proof of Proposition \ref{['pro:stability_conditions']}
  • ...and 84 more