On the Asymptotic Behavior of the Trajectories of Semigroups of Holomorphic Functions

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• 作者：Dimitrios Betsakos
• 关键词：Semigroup of holomorphic functions ; Univalent function ; Domains convex in one direction ; Harmonic measure ; 30D05 ; 37F99 ; 30C45 ; 30C85
• 刊名：Journal of Geometric Analysis
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：26
• 期：1
• 页码：557-569
• 全文大小：415 KB
• 参考文献：1.Carathéodory, C.: Theory of Functions of a Complex Variable, vol. 1. Chelsea Publishing, New York (1960). Second english edition
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• 作者单位：Dimitrios Betsakos (1)
  
  1. Department of Mathematics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Differential Geometry
  Convex and Discrete Geometry
  Fourier Analysis
  Abstract Harmonic Analysis
  Dynamical Systems and Ergodic Theory
  Global Analysis and Analysis on Manifolds
  
• 出版者：Springer New York
• ISSN：1559-002X

文摘

Let \(\{\phi _t\}_{t\ge 0}\) be a semigroup of holomorphic self-maps of the unit disk. We assume that the Denjoy–Wolff point of the semigroup is the point 1; so 1 is the unique attractive boundary fixed point of the semigroup. We further assume that for all \(t\ge 0\), \(\phi _t^\prime (1)=1\) (angular derivative), namely the semigroup is parabolic. We disprove a conjecture of Contreras and Díaz-Madrigal on the asymptotic behavior of the trajectories \(\gamma _z(t)=\phi _t(z)\), as \(t\rightarrow +\infty \). We also prove that if the boundary of the associated planar domain is contained in a half-strip, then all the trajectories of the semigroup converge to 1 radially. Keywords Semigroup of holomorphic functions Univalent function Domains convex in one direction Harmonic measure