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

 

On the connectivity of the escaping set for complex exponential Misiurewicz parameters


Author: Xavier Jarque
Journal: Proc. Amer. Math. Soc. 139 (2011), 2057-2065
MSC (2010): Primary 37F50; Secondary 32A15, 37B10
Published electronically: November 18, 2010
MathSciNet review: 2775383
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Abstract: Let $ E_{\lambda}(z)=\lambda {\rm exp}(z), \lambda\in \mathbb{C}$, be the complex exponential family. For all functions in the family there is a unique asymptotic value at 0 (and no critical values). For a fixed $ \lambda$, the set of points in $ \mathbb{C}$ with orbit tending to infinity is called the escaping set. We prove that the escaping set of $ E_{\lambda}$ with $ \lambda$ Misiurewicz (that is, a parameter for which the orbit of the singular value is strictly preperiodic) is a connected set.


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Additional Information

Xavier Jarque
Affiliation: Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran via 585, 08007 Barcelona, Catalunya, Spain
Address at time of publication: Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Catalunya, Spain
Email: xavier.jarque@bu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10611-1
PII: S 0002-9939(2010)10611-1
Received by editor(s): August 13, 2009
Received by editor(s) in revised form: December 11, 2009, and May 31, 2010
Published electronically: November 18, 2010
Additional Notes: The author is partially supported by grants 2009SGR–792, MTM2006–05849 and MTM–2008–01486 Consolider (including a FEDER contribution).
Communicated by: Bryna R. Kra
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.