Escaping points of exponential maps

Dierk Schleicher, Johannes Zimmer

Research output: Contribution to journalArticlepeer-review

79 Scopus citations

Abstract

The points which converge to ∞ under iteration of the maps z → λ exp(z) for λ ∈ ℂ\{0} are investigated. A complete classification of such 'escaping points' is given: they are organized in the form of differentiate 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.

Original languageEnglish
Pages (from-to)380-400
Number of pages21
JournalJournal of the London Mathematical Society
Volume67
Issue number2
DOIs
StatePublished - Apr 2003
Externally publishedYes

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