Non-landing hairs in Sierpinski curve Julia sets of transcendental entire maps
Garijo, Antoni (Universitat Rovira i Virgili. Departament d'Enginyeria Informàtica i Matemàtiques)
Jarque i Ribera, Xavier (Universitat Rovira i Virgili. Departament d'Enginyeria Informàtica i Matemàtiques)
Moreno Rocha, Mónica (Centro de Investigación en Matemáticas (Guanajuato, México))

Date:	2011
Abstract:	We consider the family of transcendental entire maps given by fa (z) = a(z - (1 - a)) exp(z + a) where a is a complex parameter. Every map has a superattracting fixed point at z = -a and an asymptotic value at z = 0. For a > 1 the Julia set of fa is known to be homeomorphic to the Sierpi' nski universal curve [19], thus containing embedded copies of any one-dimensional plane continuum. In this paper we study subcontinua of the Julia set that can be defined in a combinatorial manner. In particular, we show the existence of non-landing hairs with prescribed combinatorics embedded in the Julia set for all parameters a ≥ 3. We also study the relation between non-landing hairs and the immediate basin of attraction of z = -a. Even as each non-landing hair accumulates onto the boundary of the immediate basin at a single point, its closure, nonetheless, becomes an indecomposable subcontinuum of the Julia set.
Grants:	Agència de Gestió d'Ajuts Universitaris i de Recerca 2009/SGR-792 Ministerio de Economía y Competitividad MTM200801486 Ministerio de Economía y Competitividad MTM2006-05849
Note:	Agraïments: The first and second author are both partially supported by the European network 035651-2-CODY. The third author is supported by CONACyT grant 59183, CB-2006-01.
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Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Transcendental entire maps ; Julia set ; Non-landing hairs ; Indecomposable continua
Published in:	Fundamenta Mathematicae, Vol. 214 (2011) , p. 135-160, ISSN 1730-6329

DOI: 10.4064/fm214-2-3

Postprint 32 p, 489.6 KB

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