Semigroups of Holomorphic Functions

  • 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 primary subject of our study: continuous one-parameter semigroups of holomorphic self-maps of the unit disc. We establish their main basic properties and extend to this context the Denjoy-Wolff Theorem. Then we characterize groups of automorphisms and more generally of linear fractional self-maps of the unit disc. We also briefly consider continuous semigroups of holomorphic self-maps of \(\mathbb C\) and \(\mathbb C_\infty \), proving that they reduce to groups of Möbius transformations, and we explain why a non-trivial theory of continuous semigroups of holomorphic maps only makes sense for self-maps of the unit disc.

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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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