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

Benini, Anna Miriam and Rempe-Gillen, Lasse ORCID: 0000-0001-8032-8580 A landing theorem for entire functions with bounded post-singular sets.

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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
Additional Information:	58 pages, 3 figures. V3: Corrections and clarifications throughout. In particular, a new Section 7 is introduced, which concerns separation properties of dreadlocks landing at a point
Uncontrolled Keywords:	math.DS, math.DS
Depositing User:	Symplectic Admin
Date Deposited:	03 Jun 2020 08:57
Last Modified:	18 Jan 2023 23:50
Related URLs:	Author
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3089323

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• A landing theorem for entire functions with bounded post-singular sets. (deposited 03 Jun 2020 08:57) [Currently Displayed]
  • A landing theorem for entire functions with bounded post-singular sets. (deposited 16 Sep 2020 10:17)
    • A landing theorem for entire functions with bounded post-singular sets. (deposited 02 Dec 2020 15:03)