Hausdorff dimension of exponential parameter rays and their endpoints
- Author
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Mihai Bailesteanu 1, Horia Vlad Balan 2 and Dierk Schleicher 3
- Affiliations
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1 Department of Mathematics, Malott Hall, Cornell University, Ithaca, NY 14853, USA
2 Ming Hsieh Department of Electrical Engineering, University of Southern California, Hughes Aircraft Electrical Engineering Center, Los Angeles, CA 90089-2560, USA
3 School of Engineering and Science, Jacobs University Bremen (Formerly International University Bremen), Postfach 750 561, D-28725 Bremen, Germany
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mb452@cornell.edu vlad.gm@gmail.com dierk@jacobs-university.de
- Journal
- Issue
- Citation
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Mihai Bailesteanu et al 2008 Nonlinearity 21 113
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- Abstract
We investigate the set I of parameters κ for which the singular value of z
ez + κ converges to ∞. The set I consists of uncountably many parameter rays, plus landing points of some of these rays (Förster et al 2008 Proc Am. Math. Soc. 136 at press (Preprint math.DS/0311427)). We show that the parameter rays have Hausdorff dimension 1, which implies (Qiu 1994 Acta Math. Sin. (N.S.) 10 362–8) that the ray endpoints in I alone have dimension 2. Analogous results were known for dynamical planes of exponential maps (Karpińska 1999 C. R. Acad. Sci. Paris Sér. I: Math. 328 1039–44; Schleicher and Zimmer 2003 J. Lond. Math. Soc. 67 380–400); our result shows that this also holds in parameter space.- PACS
- MSC
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37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations
- Subjects
- Dates
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Issue 1 (January 2008)
Received 23 April 2007 , in final form 17 October 2007
Published 17 December 2007
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Hausdorff dimension of exponential parameter rays and their endpoints
Mihai Bailesteanu et al 2008 Nonlinearity 21 113
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Scientific reticence and sea level rise
J E Hansen 2007 Environ. Res. Lett. 2 024002
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