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Geometric description of the classification of holomorphic semigroups


Author: Dimitrios Betsakos
Journal: Proc. Amer. Math. Soc. 144 (2016), 1595-1604
MSC (2010): Primary 30D05, 37L05, 30C45
DOI: https://doi.org/10.1090/proc/12814
Published electronically: July 8, 2015
MathSciNet review: 3451236
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Abstract: We consider parabolic semigroups $ (\phi _t)_{t\geq 0}$ of holomorphic self-maps of the unit disk $ \mathbb{D}$ with Denjoy-Wolff point $ 1$, Koenigs function $ h$ and associated planar domain $ \Omega $. We give a geometric description of the classification of such semigroups: The semigroup is of positive hyperbolic step if and only if $ \Omega $ is contained in a horizontal half-plane. Moreover, a semigroup of positive hyperbolic step has trajectories that converge to $ 1$ strongly tangentially (namely the semigroup is of finite shift) if and only if $ h$ is conformal at $ 1$.


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Additional Information

Dimitrios Betsakos
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Email: betsakos@math.auth.gr

DOI: https://doi.org/10.1090/proc/12814
Keywords: Semigroup of holomorphic functions, Denjoy-Wolff point, Koenigs function, parabolic semigroup, Abel equation, positive hyperbolic step, finite shift, angular derivative
Received by editor(s): December 2, 2014
Received by editor(s) in revised form: April 20, 2015
Published electronically: July 8, 2015
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2015 American Mathematical Society