Brushing the hairs of transcendental entire functions
Barański, Krzysztof (University of Warsaw(Poland). Institute of Mathematics)
Jarque i Ribera, Xavier (Universitat Rovira i Virgili. Departament d'Enginyeria Informàtica i Matemàtiques)
Rempe, Lasse (University of Liverpool(UK). Department of Mathematical Sciences)

Data:	2012
Resum:	Let f be a transcendental entire function of finite order in the EremenkoLyubich class B (or a finite composition of such maps), and suppose that f is hyperbolic and has a unique Fatou component. We show that the Julia set of f is a Cantor bouquet; i. e. is ambiently homeomorphic to a straight brush in the sense of Aarts and Oversteegen. In particular, we show that any two such Julia sets are ambiently homeomorphic. We also show that if f ∈ B has finite order (or is a finite composition of such maps), but is not necessarily hyperbolic with connected Fatou set, then the Julia set of f contains a Cantor bouquet. As part of our proof, we describe, for an arbitrary function f ∈ B, a natural compactification of the dynamical plane by adding a "circle of addresses" at infinity.
Ajuts:	Ministerio de Economía y Competitividad MTM2006-05849 Ministerio de Economía y Competitividad MTM2008-01486 Agència de Gestió d'Ajuts Universitaris i de Recerca 2009/SGR-792
Nota:	Agraïments: The first author is supported by Polish MNiSW Grant N N201 0234 33 and Polish MNiSW SPB-M. The third author is supported by EPSRC fellowship EP/E052851/1. All three authors are supported by the EU FP6 Marie Curie Program RTN CODY.
Drets:	Tots els drets reservats.
Llengua:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Publicat a:	Topology and its applications, Vol. 159 Núm. 8 (2012) , p. 2102-2114, ISSN 0166-8641

DOI: 10.1016/j.topol.2012.02.004

Postprint 19 p, 385.7 KB

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