Criniferous entire maps with absorbing Cantor bouquets

Leticia Pardo-Simón

doi:10.3934/dcds.2021144

Article Contents

2022, Volume 42, Issue 2: 989-1010. Doi: 10.3934/dcds.2021144

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Criniferous entire maps with absorbing Cantor bouquets

Leticia Pardo-Simón

Department of Mathematics, The University of Manchester, Manchester, M13 9PL, UK

Received: October 2020

Revised: July 2021

Early access: October 12, 2021

Published online: October 12, 2021

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Abstract

It is known that, for many transcendental entire functions in the Eremenko-Lyubich class $ \mathcal{B} $, every escaping point can eventually be connected to infinity by a curve of escaping points. When this is the case, we say that the functions are criniferous. In this paper, we extend this result to a new class of maps in $ \mathcal{B} $. Furthermore, we show that if a map belongs to this class, then its Julia set contains a Cantor bouquet; in other words, it is a subset of $ \mathbb{C} $ ambiently homeomorphic to a straight brush.

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Mathematics Subject Classification: Primary 37F10; Secondary 54H20, 30D05, 54F15.

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Figure 1. On the left, hairs of a Cantor bouquet intersecting a circle $ \partial {{\mathbb{D}}}_R $, some of them multiple times. For each hair, dashes represent points with lower potential than that of the last point that intersects $ \partial {{\mathbb{D}}}_R $. On the right, the image of the hairs to a straight brush under an ambient homeomorphism $ \psi $. $ [-Q, Q]^2 $ is a square whose boundary the hairs intersect at most once, and $ S_R: = \psi^{-1}((-Q, Q)^2) $

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Figure 2. Construction of a neighbourhood of $ z_n(\eta) $ in Claim Claim 2 by pulling back balls centred at $ f^j(z_n) $ for all $ 1\leq j\leq n $ such that $ f^j(z_n) \in \partial S_R $

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Figure 3. Proof of Proposition 8 by interpolating the maps $ \psi_g $ and $ {\varphi}_f $ using the annulus $ \mathcal{R} $ shown in orange

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