Journal of the London Mathematical Society



Notes and Papers

ESCAPING POINTS OF EXPONENTIAL MAPS


DIERK SCHLEICHER a1 and JOHANNES ZIMMER a2
a1 School of Engineering and Science, International University Bremen, Postfach 750 561, D-28725 Bremen, Germany dierk@iu-bremen.de
a2 Division of Engineering and Applied Science, Mail Stop 104-44, California Institute of Technology, Pasadena, CA 91125, USA zimmer@caltech.edu

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 Karpinska for specific choices of $\lambda$.

(Received May 24 2000)
(Revised June 11 2001)

Maths Classification

30D05; 33B10; 37B10; 37B45; 37C35; 37C45; 37C70; 37F10; 37F20; 37F35.