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ESCAPING POINTS OF EXPONENTIAL MAPS

Published online by Cambridge University Press:  24 March 2003

DIERK SCHLEICHER
Affiliation:
School of Engineering and Science, International University Bremen, Postfach 750 561, D-28725 Bremen, Germanydierk@iu-bremen.de
JOHANNES ZIMMER
Affiliation:
Division of Engineering and Applied Science, Mail Stop 104-44, California Institute of Technology, Pasadena, CA 91125, USAzimmer@caltech.edu
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Abstract

The points which converge to $\infty$ under iteration of the maps $z\mapsto\lambda\exp(z)$ for $\lambda \in \mathbb{C} \backslash \{0\}$ 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 $\lambda$.

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 $\lambda$.

Type
Notes and Papers
Copyright
The London Mathematical Society 2003

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