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Trees and hairs for some hyperbolic entire maps of finite order

Trees and hairs for some hyperbolic entire maps of finite order Let f be an entire transcendental map of finite order, such that all the singularities of f −1 are contained in a compact subset of the immediate basin B of an attracting fixed point. It is proved that there exist geometric coding trees of preimages of points from B with all branches convergent to points from $${\hat {\mathbb C}}$$ . This implies that the Riemann map onto B has radial limits everywhere. Moreover, the Julia set of f consists of disjoint curves (hairs) tending to infinity, homeomorphic to a half-line, composed of points with a given symbolic itinerary and attached to the unique point accessible from B (endpoint of the hair). These facts generalize the corresponding results for exponential maps. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

Trees and hairs for some hyperbolic entire maps of finite order

Mathematische Zeitschrift , Volume 257 (1) – Mar 2, 2007

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References (29)

Publisher
Springer Journals
Copyright
Copyright © 2007 by Springer-Verlag
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/s00209-007-0114-7
Publisher site
See Article on Publisher Site

Abstract

Let f be an entire transcendental map of finite order, such that all the singularities of f −1 are contained in a compact subset of the immediate basin B of an attracting fixed point. It is proved that there exist geometric coding trees of preimages of points from B with all branches convergent to points from $${\hat {\mathbb C}}$$ . This implies that the Riemann map onto B has radial limits everywhere. Moreover, the Julia set of f consists of disjoint curves (hairs) tending to infinity, homeomorphic to a half-line, composed of points with a given symbolic itinerary and attached to the unique point accessible from B (endpoint of the hair). These facts generalize the corresponding results for exponential maps.

Journal

Mathematische ZeitschriftSpringer Journals

Published: Mar 2, 2007

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