Annales Academię Scientiarum Fennicę
Mathematica
Volumen 32, 2007, 353-369

ON NONLANDING DYNAMIC RAYS OF EXPONENTIAL MAPS

Lasse Rempe

University of Liverpool, Department of Mathematical Sciences
Liverpool L69 7ZL, United Kingdom; l.rempe 'at' liverpool.ac.uk

Abstract. We consider the case of an exponential map E_\kappa : z \mapsto exp(z) + \kappa for which the singular value \kappa is accessible from the set of escaping points of E_\kappa. We show that there are dynamic rays of E_\kappa which do not land. In particular, there is no analog of Douady's "pinched disk model" for exponential maps whose singular value belongs to the Julia set. We also prove that the boundary of a Siegel disk U for which the singular value is accessible both from the set of escaping points and from U contains uncountably many indecomposable continua.

2000 Mathematics Subject Classification: Primary 37F10; Secondary 30D05, 54F15.

Key words: Complex dynamics, Julia set, Siegel disk, exponential map, indecomposable continuum, dynamic ray.

Reference to this article: L. Rempe: On nonlanding dynamic rays of exponential maps. Ann. Acad. Sci. Fenn. Math. 32 (2007), 353-369.

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