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Horn maps of semi-parabolic Hénon maps

Astorg Matthieu, Bianchi Fabrizio

Abstract

We prove that horn maps associated to quadratic semi-parabolic fixed points of Hénon maps, first introduced by Bedford, Smillie, and Ueda, satisfy a weak form of the Ahlfors island property. As a consequence, two natural definitions of their Julia set (the non-normality locus of the family of iterates and the closure of the set of the repelling periodic points) coincide. As another consequence, we also prove that there exist small perturbations of semi-parabolic Hénon maps for which the Hausdorff dimension of the forward Julia set $J^+$ is arbitrarily close to 4.

Horn maps of semi-parabolic Hénon maps

Abstract

We prove that horn maps associated to quadratic semi-parabolic fixed points of Hénon maps, first introduced by Bedford, Smillie, and Ueda, satisfy a weak form of the Ahlfors island property. As a consequence, two natural definitions of their Julia set (the non-normality locus of the family of iterates and the closure of the set of the repelling periodic points) coincide. As another consequence, we also prove that there exist small perturbations of semi-parabolic Hénon maps for which the Hausdorff dimension of the forward Julia set is arbitrarily close to 4.
Paper Structure (15 sections, 23 theorems, 22 equations)

This paper contains 15 sections, 23 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Island properties and consequences
  3. Ahlfors island maps and finite type maps
  4. Julia sets
  5. A consequence of the small island property
  6. Preliminaries on Hénon and horizontal-like maps
  7. Filtration property and induced horizontal-like map
  8. The Green currents $T^+$ and $T^-$
  9. The equilibrium measure and Pesin boxes
  10. Semi-parabolic dynamics and horn maps
  11. Proof of Theorem \ref{['th:islandp']}
  12. Almost maximal dimension for $J^+$ and Theorem \ref{['th:maxdim']}
  13. Preliminaries and McMullen's result
  14. Large hyperbolic sets from the small island property
  15. Proof of Theorem \ref{['th:maxdim']}

Key Result

Theorem 1.2

Let $f$ be a dissipative semi-parabolic Hénon map as above. The horn map $h_f$ has the small island property as in Definition d:small-island.

Theorems & Definitions (48)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5: Definition of the Julia set as non-normality locus
  • Definition 2.6: Definition of the Julia set by means of repelling periodic points
  • ...and 38 more