Contact points and fractional singularities for semigroups of\n holomorphic self-maps in the unit disc

2013/09/11 by Filippo Bracci, Pavel Gumenyuk, Bracci, Filippo +1
Mathematics · #30C55 #30C80 #30D05 #30D40 #37C10 #37C25 #37F75 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory #Mathematical Dynamics and Fractals #Primary 30C35 #Secondary 30C45

paper · pdf · doi:10.48550/arxiv.1309.2813

openalex publication_date 2013/09/11 · openalex created_date 2022/08/21 · openalex updated_date 2026/07/28

Abstract

We study boundary singularities which can appear for infinitesimal generators\nof one-parameter semigroups of holomorphic self-maps in the unit disc. We\nintroduce "regular" fractional singularities and characterize them in terms of\nthe behavior of the associated semigroups and Koenigs functions. We also\nprovide necessary and sufficient geometric criteria on the shape of the image\nof the Koenigs function for having such singularities. In order to do this, we\nstudy contact points of semigroups and prove that any contact (not fixed) point\nof a one-parameter semigroup corresponds to a maximal arc on the boundary to\nwhich the associated infinitesimal generator extends holomorphically as a\nvector field tangent to this arc.\n

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