Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Boundaries of hyperbolic and simply parabolic Baker domains

Anna Jové

Abstract

We study the boundaries of non-univalent simply connected Baker domains of transcendental maps (both entire and meromorphic), of hyperbolic and simply parabolic type. We prove non-ergodicity and non-recurrence for the boundary map, and additional properties concerning the Julia set and the set of singularities of the associated inner function, and the topology and the dynamics on the boundary of the Baker domain. In particular, we prove the existence of points on the boundary whose orbit does not converge to infinity through the dynamical access, in the sense of Carathéodory. Finally, under mild conditions on the postsingular set, we prove the existence of periodic points on the boundary of such Baker domains.

Boundaries of hyperbolic and simply parabolic Baker domains

Abstract

We study the boundaries of non-univalent simply connected Baker domains of transcendental maps (both entire and meromorphic), of hyperbolic and simply parabolic type. We prove non-ergodicity and non-recurrence for the boundary map, and additional properties concerning the Julia set and the set of singularities of the associated inner function, and the topology and the dynamics on the boundary of the Baker domain. In particular, we prove the existence of points on the boundary whose orbit does not converge to infinity through the dynamical access, in the sense of Carathéodory. Finally, under mild conditions on the postsingular set, we prove the existence of periodic points on the boundary of such Baker domains.

Paper Structure

This paper contains 12 sections, 20 theorems, 87 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Inner function associated with a hyperbolic or simply parabolic Baker domain
  3. Iteration of inner functions associated with Baker domains
  4. Inverse branches for inner functions at points in the unit circle
  5. Boundary behaviour of the Riemann map
  6. Ergodic properties of $f\colon \partial U\to\partial U$. Proof of \ref{['teo:A']}
  7. Hyperbolic and simply parabolic Baker domains of entire functions. Proof of \ref{['teo:B']} and \ref{['prop:C']}
  8. The Carathéodory set
  9. The Hausdorff dimension of $\mathcal{J}(g)$
  10. Proof of \ref{['prop:C']}
  11. Proof of \ref{['teo:B']}
  12. Boundary dynamics. Proof of \ref{['teo:D']}

Key Result

Theorem 2.1

(Cowen's classification of self-maps of $\mathbb{D}$, cowen) Let $g$ be a holomorphic self-map of $\mathbb{D}$ with Denjoy-Wolff point $p\in\partial \mathbb{D}$. Then, there exists a simply connected domain $V \subset\mathbb{D}$, a domain $\Omega$ equal to $\mathbb{C}$ or $\mathbb{H}=\left\lbrace \t Moreover, $T$ and $\Omega$ depend only on the map $g$, not on the absorbing domain $V$. In fact (up

Figures (11)

  • Figure 1.1: Possible dynamics around the Denjoy-Wolff point, when it is not a singularity.
  • Figure 2.1: The different types of convergence to the Denjoy-Wolff point.
  • Figure 2.2: Consider the inner function $h\colon\mathbb{H}\to\mathbb{H}$, with Denjoy-Wolff point $\infty$. Assuming that there are no singular values in some crosscut neighbourhood of $x\in\mathbb{R}$ (grey), inverse branches are well-defined in a disk around $x$ (light grey), and one can control the distortion on the radial segment in terms of Stolz angles (purple). The results transfer straight-forward to the unit circle by means of the Möbius transformation $M$.
  • Figure 3.1: Set-up of the proof of non-recurrence: schematic representation the Riemann map $\varphi\colon\mathbb{D}\to U$, together with the choice of $\xi_1, \xi_2\in\partial \mathbb{D}$.
  • Figure 4.1: Dynamical plane of $f(z)=z+e^{-z}$, with the doubly parabolic Baker domain $U_0$ (orange). The Riemann map $\varphi\colon\mathbb{D}\to U_0$ is depicted, together with the inner function. Note that 1 is the Denjoy-Wolff point, and it is repelling when restricted to $\partial\mathbb{D}$. Crosscut neighbourhoods at the Denjoy-Wolff point are indicated, as well as their image in the dynamical plane. By definition, the Carathéodory set consists of those points on $\partial U$ whose orbit eventually enters the image of every crosscut neighbourhood of 1. By the dynamics of $g$, one deduces that the Carathéodory set is $Cl(\varphi, 1)\cap \mathbb{C}$ and its iterated preimages.
  • ...and 6 more figures

Theorems & Definitions (38)

  • Definition
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • Proposition 2.8
  • ...and 28 more