Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the rates of convergence of orbits in semigroups of holomorphic functions

Dimitrios Betsakos, Francisco J. Cruz-Zamorano, Konstantinos Zarvalis

Abstract

Let $(φ_t)$ be a continuous semigroup of holomorphic self-maps of the unit disk $\mathbb{D}$ with Denjoy-Wolff point $τ\in\overline{\mathbb{D}}$. We study the rate of convergence of the forward orbits of $(φ_t)$ to the Denjoy-Wolff point by finding explicit bounds for the quantity $|φ_t(z)-τ|$, $z\in\overline{\mathbb{D}}$, $t > 0$. We further discuss the corresponding rate of convergence for the backward orbits of $(φ_t)$.

On the rates of convergence of orbits in semigroups of holomorphic functions

Abstract

Let be a continuous semigroup of holomorphic self-maps of the unit disk with Denjoy-Wolff point . We study the rate of convergence of the forward orbits of to the Denjoy-Wolff point by finding explicit bounds for the quantity , , . We further discuss the corresponding rate of convergence for the backward orbits of .

Paper Structure

This paper contains 16 sections, 14 theorems, 174 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Statement of Main Results
  3. Auxiliary Information
  4. Semigroups in the Unit Disk
  5. Backward Orbits
  6. Infinitesimal Generators
  7. Hyperbolic Geometry
  8. Harmonic Measure
  9. Extremal Length and Extremal Distance
  10. Forward Orbits in Non-elliptic Semigroups
  11. Upper Bounds
  12. Lower Bound
  13. Backward Orbits in Non-elliptic Semigroups
  14. Regular Backward Orbits
  15. Non-regular Backward Orbits
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $(\phi_t)$ be a non-elliptic semigroup in $\mathbb{D}$ with Denjoy-Wolff point $\tau$ and Koenigs function $h$. Suppose that $\Delta$ is a petal of $(\phi_t)$ and that all backward orbits contained in $\Delta$ converge to $\sigma\in\partial\mathbb{D}$. Then, for every $z \in \Delta$ and all $t >

Figures (4)

  • Figure 1: Construction in the proof of Theorem \ref{['thm:forward']}(a).
  • Figure 2: (A) Construction in the proof of Theorem \ref{['thm:forward']}(b), (B) Construction in the proof of Theorem \ref{['thm:forward']}(c).
  • Figure 3: Construction in the proof of Theorem \ref{['thm:lower']}.
  • Figure 4: Construction in the proof of Theorem \ref{['thm:backward']}(c).

Theorems & Definitions (33)

  • Theorem 2.1: Non-elliptic semigroups, regular backward orbits
  • Theorem 2.2: Non-elliptic semigroups, non-regular backward orbits
  • Theorem 2.3: Non-elliptic semigroups, forward orbits, upper bounds
  • Theorem 2.4: Non-elliptic semigroups, forward orbits, lower bound
  • Theorem 2.5: Elliptic semigroups, all orbits
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma 3.4
  • Example 3.5
  • ...and 23 more