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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Poincaré functions with spiders' webs


Authors: Helena Mihaljević-Brandt and Jörn Peter
Journal: Proc. Amer. Math. Soc. 140 (2012), 3193-3205
MSC (2010): Primary 30D05; Secondary 37F10, 30D15, 37F45
Published electronically: January 20, 2012
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Abstract: For a polynomial $ p$ with a repelling fixed point $ z_0$, we consider Poincaré functions of $ p$ at $ z_0$, i.e. entire functions $ \mathfrak{L}$ which satisfy $ \mathfrak{L}(0)=z_0$ and $ p(\mathfrak{L}(z))=\mathfrak{L}( p'(z_0) \cdot z)$ for all $ z\in \mathbb{C}$. We show that if the component of the Julia set of $ p$ that contains $ z_0$ equals $ \{z_0\}$, then the (fast) escaping set of $ \mathfrak{L}$ is a spider's web; in particular, it is connected. More precisely, we classify all linearizers of polynomials with regard to the spider's web structure of the set of all points which escape faster than the iterates of the maximum modulus function at a sufficiently large point $ R$.


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Additional Information

Helena Mihaljević-Brandt
Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, 24118 Kiel, Germany
Email: helenam@math.uni-kiel.de

Jörn Peter
Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, 24118 Kiel, Germany
Email: peter@math.uni-kiel.de

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11164-5
PII: S 0002-9939(2012)11164-5
Received by editor(s): September 28, 2010
Received by editor(s) in revised form: February 16, 2011, and March 28, 2011
Published electronically: January 20, 2012
Additional Notes: The second author has been supported by the Deutsche Forschungsgemeinschaft, Be 1508/7-1. He was also partially supported by the EU Research Training Network Cody.
Communicated by: Mario Bonk
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.