A landing theorem for entire functions with bounded post-singular sets



Benini, Anna Miriam and Rempe-Gillen, Lasse
(2020) A landing theorem for entire functions with bounded post-singular sets. Geometric and Functional Analysis, 30 (6). 1465 - 1530.

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Abstract

The Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue of this theorem for an entire function with bounded postsingular set. If the function has finite order of growth, then it is known that the escaping set contains certain curves called "periodic hairs"; we show that every periodic hair lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic hair. For a postsingularly bounded entire function of infinite order, such hairs may not exist. Therefore we introduce certain dynamically natural connected sets, called "dreadlocks". We show that every periodic dreadlock lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic dreadlock. More generally, we prove that every point of a hyperbolic set is the landing point of a dreadlock.

Item Type: Article
Depositing User: Symplectic Admin
Date Deposited: 02 Dec 2020 15:03
Last Modified: 12 Nov 2021 12:23
DOI: 10.1007/s00039-020-00551-3
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3108903

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