Mathematische Zeitschrift

, Volume 292, Issue 1–2, pp 343–359 | Cite as

Nowhere differentiable hairs for entire maps

  • Patrick ComdührEmail author


Devaney and Krych (Ergod Theory Dyn Syst 4:35–52, 1984) showed that for the exponential family \(\lambda e^z\), where \(0<\lambda <1/e\), the Julia set consists of uncountably many pairwise disjoint simple curves tending to \(\infty \), which they called hairs. Viana proved that these hairs are smooth. Barański as well as Rottenfußer, Rückert, Rempe and Schleicher gave analogues of the result of Devaney and Krych for more general classes of functions. In contrast to Viana’s result we construct in this article an entire function, where the Julia set consists of hairs, which are nowhere differentiable.


Exponential map Eremenko–Lyubich class Complex dynamics Hair External ray Differentiability 

Mathematics Subject Classification

Primary 30D05 Secondary 37F10 30C65 



I would like to thank Walter Bergweiler, Lasse Rempe-Gillen, Dan Nicks and the referee for valuable suggestions.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematisches SeminarChristian-Albrechts-Universität zu KielKielGermany

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