Published Paper
Inserted: 1 apr 2024
Last Updated: 1 jul 2025
Journal: Math. Annalen
Volume: 392
Pages: 837-860
Year: 2025
Abstract:
We prove that horn maps associated to quadratic semi-parabolic fixed points of H\'enon 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 H\'enon maps for which the Hausdorff dimension of the forward Julia set $J^+$ is arbitrarily close to 4.