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February  2022, 42(2): 989-1010. doi: 10.3934/dcds.2021144

Criniferous entire maps with absorbing Cantor bouquets

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

Received  October 2020 Revised  July 2021 Published  February 2022 Early access  October 2021

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.

Citation: Leticia Pardo-Simón. Criniferous entire maps with absorbing Cantor bouquets. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 989-1010. doi: 10.3934/dcds.2021144
References:
[1]

J. Aarts and L. Oversteegen, The geometry of Julia sets, Trans. Amer. Math. Soc., 338 (1993), 897-918.  doi: 10.1090/S0002-9947-1993-1182980-3.

[2]

K. Barański, Trees and hairs for some hyperbolic entire maps of finite order, Math. Z., 257 (2007), 33-59.  doi: 10.1007/s00209-007-0114-7.

[3]

K. BarańskiX. Jarque and L. Rempe, Brushing the hairs of transcendental entire functions, Topology Appl., 159 (2012), 2102-2114.  doi: 10.1016/j.topol.2012.02.004.

[4]

A. M. Benini and N. Fagella, Singular values and non-repelling cycles for entire transcendental maps, Indiana Univ. Math. J., 69 (2020), 1543-1558.  doi: 10.1512/iumj.2020.69.8000.

[5]

A. Benini and L. Rempe, A landing theorem for entire functions with bounded post-singular sets, Geom. Funct. Anal., 30 (2020), 1465-1530.  doi: 10.1007/s00039-020-00551-3.

[6]

A. Douady and J. H. Hubbard, Étude dynamique des polynomes complexes, Mathématiques d'Orsay [Mathematical Publications of Orsay], Université de Paris-Sud, Département de Mathématiques, Orsay, (1984), 75pp.

[7]

A. E. Erëmenko, On the iteration of entire functions, Banach Center Publications, 23 (1989), 339-345. 

[8]

A. Erëmenko and M. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble), 42 (1992), 989-1020.  doi: 10.5802/aif.1318.

[9]

L. R. Goldberg and L. Keen, A finiteness theorem for a dynamical class of entire functions, Ergodic Theory Dynam. Systems, 6 (1986), 183-192.  doi: 10.1017/S0143385700003394.

[10]

O. Lehto, An extension theorem for quasiconformal mappings, Proc. London Math. Soc., 14a (1965), 187-190.  doi: 10.1112/plms/s3-14A.1.187.

[11]

H. Mihaljević-Brandt, Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds, Trans. Amer. Math. Soc., 364 (2012), 4053-4083.  doi: 10.1090/S0002-9947-2012-05541-3.

[12]

L. Pardo Simón, Dynamics of transcendental entire functions with escaping singular orbits, PhD thesis, University of Liverpool.

[13]

L. Pardo-Simón, Splitting hairs with transcendental entire functions, Preprint, arXiv: 1905.03778v3.

[14]

L. Pardo-Simón, Topological dynamics of cosine maps, Preprint, arXiv: 2003.07250.

[15]

L. Pardo-Simón, Orbifold expansion and entire functions with bounded Fatou components, Ergodic Theory and Dynamical Systems, 1–40.

[16]

L. Rempe, Rigidity of escaping dynamics for transcendental entire functions, Acta Math., 203 (2009), 235-267.  doi: 10.1007/s11511-009-0042-y.

[17]

L. Rempe-Gillen, Arc-like continua, Julia sets of entire functions, and remenko's Conjecture, Preprint, arXiv: 1610.06278v3.

[18]

G. RottenfußerJ. RückertL. Rempe and D. Schleicher, Dynamic rays of bounded-type entire functions, Ann. of Math. (2), 173 (2011), 77-125.  doi: 10.4007/annals.2011.173.1.3.

[19]

G. Rottenfußer and D. Schleicher, Escaping points of the cosine family, Transcendental Dynamics and Complex Analysis, Cambridge University Press, 348 (2008), 396–424. doi: 10.1017/CBO9780511735233.016.

[20]

D. Schleicher and J. Zimmer, Escaping points of exponential maps, J. London Math. Soc., 67 (2003), 380-400.  doi: 10.1112/S0024610702003897.

show all references

References:
[1]

J. Aarts and L. Oversteegen, The geometry of Julia sets, Trans. Amer. Math. Soc., 338 (1993), 897-918.  doi: 10.1090/S0002-9947-1993-1182980-3.

[2]

K. Barański, Trees and hairs for some hyperbolic entire maps of finite order, Math. Z., 257 (2007), 33-59.  doi: 10.1007/s00209-007-0114-7.

[3]

K. BarańskiX. Jarque and L. Rempe, Brushing the hairs of transcendental entire functions, Topology Appl., 159 (2012), 2102-2114.  doi: 10.1016/j.topol.2012.02.004.

[4]

A. M. Benini and N. Fagella, Singular values and non-repelling cycles for entire transcendental maps, Indiana Univ. Math. J., 69 (2020), 1543-1558.  doi: 10.1512/iumj.2020.69.8000.

[5]

A. Benini and L. Rempe, A landing theorem for entire functions with bounded post-singular sets, Geom. Funct. Anal., 30 (2020), 1465-1530.  doi: 10.1007/s00039-020-00551-3.

[6]

A. Douady and J. H. Hubbard, Étude dynamique des polynomes complexes, Mathématiques d'Orsay [Mathematical Publications of Orsay], Université de Paris-Sud, Département de Mathématiques, Orsay, (1984), 75pp.

[7]

A. E. Erëmenko, On the iteration of entire functions, Banach Center Publications, 23 (1989), 339-345. 

[8]

A. Erëmenko and M. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble), 42 (1992), 989-1020.  doi: 10.5802/aif.1318.

[9]

L. R. Goldberg and L. Keen, A finiteness theorem for a dynamical class of entire functions, Ergodic Theory Dynam. Systems, 6 (1986), 183-192.  doi: 10.1017/S0143385700003394.

[10]

O. Lehto, An extension theorem for quasiconformal mappings, Proc. London Math. Soc., 14a (1965), 187-190.  doi: 10.1112/plms/s3-14A.1.187.

[11]

H. Mihaljević-Brandt, Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds, Trans. Amer. Math. Soc., 364 (2012), 4053-4083.  doi: 10.1090/S0002-9947-2012-05541-3.

[12]

L. Pardo Simón, Dynamics of transcendental entire functions with escaping singular orbits, PhD thesis, University of Liverpool.

[13]

L. Pardo-Simón, Splitting hairs with transcendental entire functions, Preprint, arXiv: 1905.03778v3.

[14]

L. Pardo-Simón, Topological dynamics of cosine maps, Preprint, arXiv: 2003.07250.

[15]

L. Pardo-Simón, Orbifold expansion and entire functions with bounded Fatou components, Ergodic Theory and Dynamical Systems, 1–40.

[16]

L. Rempe, Rigidity of escaping dynamics for transcendental entire functions, Acta Math., 203 (2009), 235-267.  doi: 10.1007/s11511-009-0042-y.

[17]

L. Rempe-Gillen, Arc-like continua, Julia sets of entire functions, and remenko's Conjecture, Preprint, arXiv: 1610.06278v3.

[18]

G. RottenfußerJ. RückertL. Rempe and D. Schleicher, Dynamic rays of bounded-type entire functions, Ann. of Math. (2), 173 (2011), 77-125.  doi: 10.4007/annals.2011.173.1.3.

[19]

G. Rottenfußer and D. Schleicher, Escaping points of the cosine family, Transcendental Dynamics and Complex Analysis, Cambridge University Press, 348 (2008), 396–424. doi: 10.1017/CBO9780511735233.016.

[20]

D. Schleicher and J. Zimmer, Escaping points of exponential maps, J. London Math. Soc., 67 (2003), 380-400.  doi: 10.1112/S0024610702003897.

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) $
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 $
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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