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Escaping Points of Exponential Maps

Schleicher, Dierk and Zimmer, Johannes (2003) Escaping Points of Exponential Maps. Journal of the London Mathematical Society, 67 (2). pp. 380-400. ISSN 0024-6107. doi:10.1112/S0024610702003897.

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The points which converge to ∞ under iteration of the maps z↦λexp(z) for λ ∈ C/{0} are investigated. A complete classification of such ‘escaping points’ is given: they are organized in the form of differentiable curves called rays which are diffeomorphic to open intervals, together with the endpoints of certain (but not all) of these rays. Every escaping point is either on a ray or the endpoint (landing point) of a ray. This answers a special case of a question of Eremenko. The combinatorics of occurring rays, and which of them land at escaping points, are described exactly. It turns out that this answer does not depend on the parameter λ. It is also shown that the union of all the rays has Hausdorff dimension 1, while the endpoints alone have Hausdorff dimension 2. This generalizes results of Karpińska for specific choices of λ.

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Additional Information:© 2003 London Mathematical Society. Received May 24, 2000; Revision received June 11, 2001. This project was inspired by discussions with Bogusia Karpińska and Misha Lyubich at a Euroconference in Crete organized by Shaun Bullett, Adrien Douady and Christos Kourouniotis. We also thank Bob Devaney, Núria Fagella, John Hubbard and Lasse Rempe for interesting discussions. We gratefully acknowledge support and encouragement by John Milnor and the Institute for Mathematical Sciences in Stony Brook. Much of this work was carried out while we held positions at the Ludwig-Maximilians-Universität München and the Technische Universität München, respectively.
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Stony Brook Institute for Mathematical SciencesUNSPECIFIED
Issue or Number:2
Classification Code:2000 Mathematics Subject Classication: 30D05, 33B10, 37B10, 37B45, 37C35, 37C45, 37C70, 37F10, 37F20, 37F35
Record Number:CaltechAUTHORS:20111019-140919972
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:27312
Deposited On:20 Oct 2011 20:37
Last Modified:09 Nov 2021 16:47

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