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

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ID: 115886
2018
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Ranked #11 of 17 articles by views in annali di matematica pura ed applicata (1923 -)

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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 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.
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Authors Filippo Bracci, Manuel D. Contreras, Santiago Díaz-Madrigal, Hervé Gaussier
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Journal annali di matematica pura ed applicata (1923 -)
Year 2018
DOI
doi:10.1007/s10231-018-0783-3
URL
https://link.springer.com/article/10.1007/s10231-018-0783-3
Keywords
mathematics general

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