Backward orbits and petals of semigroups of holomorphic self-maps of the unit disc

  • Filippo Bracci
  • Manuel D. Contreras
  • Santiago Díaz-Madrigal
  • Hervé Gaussier


We study the backward invariant set of one-parameter semigroups of holomorphic self-maps of the unit disc. Such a set is foliated in maximal invariant curves, and its open connected components are petals, which are, in fact, images of Poggi-Corradini’s type pre-models. Hyperbolic petals are in one-to-one correspondence with repelling fixed points, while only parabolic semigroups can have parabolic petals. Petals have locally connected boundaries, and except a very particular case, they are indeed Jordan domains. The boundary of a petal contains the Denjoy–Wolff point, and except such a fixed point, the closure of a petal contains either no other boundary fixed points or a unique repelling fixed point. We also describe petals in terms of geometric and analytic behavior of Koenigs functions using divergence rate and universality of models. Moreover, we construct a semigroup having a repelling fixed point in such a way that the intertwining map of the pre-model is not regular.


Semigroups of holomorphic functions Backward orbits Petals Koenigs function Holomorphic models 

Mathematics Subject Classification

Primary 37C10 30C35 Secondary 30D05 30C80 37F99 37C25 


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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomeItaly
  2. 2.Camino de los Descubrimientos, s/n Departamento de Matemática Aplicada II and IMUSUniversidad de SevillaSevilleSpain
  3. 3.CNRS, IFUniv. Grenoble AlpesGrenobleFrance

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