A two-dimensional polynomial mapping with a wandering Fatou component

Abstract

We show that there exist polynomial endomorphisms of $\mathbb{C}^2$, possessing a wandering Fatou component. These mappings are polynomial skew-products, and can be chosen to extend holomorphically of $\mathbb{P}^2(\mathbb{C})$. We also find real examples with wandering domains in $\mathbb{R}^2$. The proof relies on parabolic implosion techniques and is based on an original idea of M. Lyubich.

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      doi = {10.2307/1971308},
      zblnumber = {0589.30022},
      }
  • [trucco] Go to document E. Trucco, "Wandering Fatou components and algebraic Julia sets," Bull. Soc. Math. France, vol. 142, iss. 3, pp. 411-464, 2014.
    @article{trucco, mrkey = {3295719},
      author = {Trucco, Eugenio},
      title = {Wandering {F}atou components and algebraic {J}ulia sets},
      journal = {Bull. Soc. Math. France},
      fjournal = {Bulletin de la Société Mathématique de France},
      volume = {142},
      year = {2014},
      number = {3},
      pages = {411--464},
      issn = {0037-9484},
      mrclass = {37P40 (37F50 37P05 37P50)},
      mrnumber = {3295719},
      mrreviewer = {Y{û}suke Okuyama},
      zblnumber = {06412749},
      URL = {http://smf4.emath.fr/Publications/Bulletin/142/html/smf_bull_142_411-464.php},
     }

Authors

Matthieu Astorg

Institut de Mathématiques de Toulouse, Université de Toulouse, CNRS, Toulouse, France

Xavier Buff

Institut de Mathématiques de Toulouse, Université de Toulouse, CNRS, Toulouse, France

Romain Dujardin

Laboratoire d'Analyse et de Mathématiques Appliquées, Université Paris-Est Marne-la-Vallée, Champs-sur-Marne, France

Han Peters

KdV Institute for Mathematics University of Amsterdam, The Netherlands

Jasmin Raissy

Institut de Mathématiques de Toulouse, Université de Toulouse, CNRS, Toulouse, France