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Mathematics > Complex Variables

arXiv:1804.09744 (math)
[Submitted on 25 Apr 2018]

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

Authors:Filippo Bracci, Manuel D. Contreras, Santiago Díaz-Madrigal, Hervé Gaussier
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Abstract: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 point or a unique repelling fixed point. We also describe petals in terms of geometric and analytic behavior of Königs 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.
Comments: 35 pages
Subjects: Complex Variables (math.CV); Dynamical Systems (math.DS)
Cite as: arXiv:1804.09744 [math.CV]
  (or arXiv:1804.09744v1 [math.CV] for this version)
  https://doi.org/10.48550/arXiv.1804.09744

Submission history

From: Filippo Bracci [view email]
[v1] Wed, 25 Apr 2018 18:27:41 UTC (31 KB)
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