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Harmonic Measures, Green Potentials and Semigroups of Holomorphic Functions

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Abstract

Let K be a compact subset of the unit disk \(\mathbb {D}\). We examine the asymptotic behavior of its trajectory under a semigroup of holomorphic self-maps (ϕt)t≥ 0 of \(\mathbb {D}\). More specifically, we obtain results concerning its geometric characteristics such as hyperbolic area, hyperbolic diameter, as well as potential theoretic quantities. Those are the harmonic measure of ϕt(K), its equilibrium measure and its Green equilibrium potential.

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Acknowledgments

I would like to thank Prof. D. Betsakos, my thesis advisor, for his advice and assistance during the preparation of this work.

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Correspondence to Maria Kourou.

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Kourou, M. Harmonic Measures, Green Potentials and Semigroups of Holomorphic Functions. Potential Anal 52, 301–319 (2020). https://doi.org/10.1007/s11118-018-9748-9

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Keywords

  • Semigroup of holomorphic functions
  • Harmonic measure
  • Green potential
  • Koenigs function
  • Equilibrium measure
  • Hyperbolic metric

Mathematics Subject Classification (2010)

  • Primary 31A15, 30D05, 30C85
  • Secondary 30C20, 47D06