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The angular derivative problem for petals of one-parameter semigroups in the unit disk

Pavel Gumenyuk, Maria Kourou, Oliver Roth

Abstract

We study the angular derivative problem for petals of one-parameter semigroups of holomorphic self-maps of the unit disk. For hyperbolic petals we prove a necessary and sufficient condition for the conformality of the petal in terms of the intrinsic hyperbolic geometry of the petal and the backward dynamics of the semigroup. For parabolic petals we characterize conformality of the petal in terms of the asymptotic behaviour of the Koenigs function at the Denjoy-Wolff point.

The angular derivative problem for petals of one-parameter semigroups in the unit disk

Abstract

We study the angular derivative problem for petals of one-parameter semigroups of holomorphic self-maps of the unit disk. For hyperbolic petals we prove a necessary and sufficient condition for the conformality of the petal in terms of the intrinsic hyperbolic geometry of the petal and the backward dynamics of the semigroup. For parabolic petals we characterize conformality of the petal in terms of the asymptotic behaviour of the Koenigs function at the Denjoy-Wolff point.
Paper Structure (23 sections, 15 theorems, 118 equations)

This paper contains 23 sections, 15 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. One-parameter semigroups in the unit disk.
  4. Holomorphic models and Koenigs function
  5. Backward orbits, invariant petals, and pre-models
  6. Conformality of a domain at a boundary point
  7. Auxiliary results
  8. Strong Markov property for the Green's function
  9. An integral criterion for conformality of domains starlike at infinity
  10. A lemma on convergence of Riemann mappings on the boundary
  11. Proof of the main result
  12. Reformulation of the problem
  13. Proof of the if-part of Theorem \ref{['thm:main2']}: condition \ref{['EQ_int-mainthrm2-comvergence']} implies conformality
  14. Necessity of condition \ref{['EQ_int-mainthrm2-comvergence']}
  15. Completion of the proof of Theorem \ref{['thm:main2']}: locally uniform convergence of (\ref{['EQ_int-mainthrm2-comvergence']})
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\Delta$ be a hyperbolic petal of a non-trivial one-parameter semigroup $(\phi_t)$ in the unit disk, and let ${z_0\in\Delta}$. Then the petal $\Delta$ is conformal at its $\alpha$-point $\sigma$ if and only if In this case the integral in EQ_int-mainthrm1-comvergence converges for every $z_0\in \Delta$, and in fact locally uniformly in $\Delta$.

Theorems & Definitions (58)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • ...and 48 more