Poles of the Infinitesimal Generators

  • Filippo BracciEmail author
  • Manuel D. Contreras
  • Santiago Díaz-Madrigal
Part of the Springer Monographs in Mathematics book series (SMM)


In this chapter, we introduce the notion of regular (boundary) poles for infinitesimal generators of semigroups. We characterize such regular poles in terms of \(\beta \)-points (i.e. pre-images of values with a positive (Carleson-Makarov) \(\beta \)-numbers) of the associated semigroup and of the associated Koenigs function. We also define a natural duality operation in the cone of infinitesimal generators and show that the regular poles of an infinitesimal generator correspond to the regular critical points of the dual generator. Finally we apply such a construction to study radial multi-slits and give an example of a non-isolated radial slit semigroup whose tip has not a positive (Carleson-Makarov) \(\beta \)-number.

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Filippo Bracci
    • 1
    Email author
  • Manuel D. Contreras
    • 2
  • Santiago Díaz-Madrigal
    • 2
  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly
  2. 2.Departamento de Matemática Aplicada II and IMUSUniversidad de SevillaSevillaSpain

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