Abstract

We describe the structure of “substantially dissipative” complex Hénon maps admitting a dominated splitting on the Julia set. We prove that the Fatou set consists of only finitely many components, each either attracting or parabolic periodic. In particular, there are no wandering components and no rotation domains. Moreover, we show that $J = J^\\star$ and the dynamics on $J$ is hyperbolic away from parabolic cycles.

Highlights

  • Complex Henon maps are polynomial automorphisms of C2 with non-trivial dynamical behavior, f : (x, y) → (p(x) − by, x), where deg p ≥ 2, b = Jac f = 0.For a small Jacobian b, it can be viewed as a perturbation of the one-dimensional polynomial p : C → C

  • Dynamics of 1D polynomials on the Fatou set is fully understood, due to the classical work of Fatou, Julia and Siegel, supplemented with Sullivan’s No Wandering Domains Theorem from the early ’80s [Su85]. This direction of research for Henon maps was initiated by Bedford and Smillie in the early ’90’s. They gave a description of the dynamics on “recurrent” periodic Fatou components [BS91b]

  • The Henon counterpart of this result was established by Bedford and Smillie in the ’90s, resulting in a complete description of the dynamics on the Fatou set for this class [BS91a]: a hyperbolic Henon map has only finitely many Fatou components, each of which is an attracting basin

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Summary

Introduction

Dynamics of 1D polynomials on the Fatou set is fully understood, due to the classical work of Fatou, Julia and Siegel, supplemented with Sullivan’s No Wandering Domains Theorem from the early ’80s [Su85] This direction of research for Henon maps was initiated by Bedford and Smillie in the early ’90’s. The Henon counterpart of this result was established by Bedford and Smillie in the ’90s, resulting in a complete description of the dynamics on the Fatou set for this class [BS91a]: a hyperbolic Henon map has only finitely many Fatou components, each of which is an attracting basin In this case, the Julia set J is the closure of saddles: J = J∗. Non-hyperbolic complex Henon maps admitting a dominated splitting have been constructed by Radu and Tanase [RT14] These examples are perturbations of 1D parabolic polynomials.

The one-dimensional argument
Background and preliminaries
Dominated splitting
Dynamical lamination and its extensions
Uniformization of wandering components
Horizontal lamination and degcrit
Final preparations
Diameter and degree bounds - proof for Henon maps
10. Consequences
Full Text
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