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<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2024_3085_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(J^+\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>J</mi> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation> is arbitrarily close to 4.</p>

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Horn maps of semi-parabolic Hénon maps

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年2月24日

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  • Matthieu Astorg,
  • Fabrizio Bianchi

摘要

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 \(J^+\) J + is arbitrarily close to 4.