Advertisement

Mathematische Zeitschrift

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

Nowhere differentiable hairs for entire maps

  • Patrick ComdührEmail author
Article
  • 26 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification

Primary 30D05 Secondary 37F10 30C65 

Notes

Acknowledgements

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

References

  1. 1.
    Barański, K.: Trees and hairs for some hyperbolic entire maps of finite order. Math. Z. 257(1), 33–59 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beardon, A.F.: Iteration of Rational Functions: Complex Analytic Dynamical Systems. Graduate Texts in Mathematics, vol. 132. Springer, New York (1991)zbMATHGoogle Scholar
  3. 3.
    Bergweiler, W.: Iteration of meromorphic functions. Bull. Am. Math. Soc. (N. S.) 29, 151–188 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barański, K., Jarque, X., Rempe, L.: Brushing the hairs of transcendental entire functions. Topol. Appl. 159(8), 2102–2114 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Comdühr, P.: On the differentiability of hairs for Zorich maps. Ergod. Theory Dyn. Syst. (2017).  https://doi.org/10.1017/etds.2017.104
  6. 6.
    Devaney, R.L., Krych, M.: Dynamics of exp(z). Ergod. Theory Dyn. Syst. 4, 35–52 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Duren, P.: Harmonic Mappings in the Plane. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  8. 8.
    Eremenko, A.È., Lyubich, M.Y.: Dynamical properties of some classes of entire functions. Ann. Inst. Fourier (Grenoble) 42, 989–1020 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fatou, P.: Sur les solutions uniformes de certaines équations fonctionelles. In: Dynamics in One Complex Variable, vol. 143, pp 546–548. C. R. Acad. Sci, Paris (1906)Google Scholar
  10. 10.
    Goldberg, A.A., Ostrovskii, I.V.: Value Distribution of Meromorphic Functions. Translations of Mathematical Monographs, vol. 236. American Mathematical Society, Providence (2008)zbMATHGoogle Scholar
  11. 11.
    Mihaljević-Brandt, H.: Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds. Trans. Am. Math. Soc. 364(8), 4053–4083 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Milnor, J.: Dynamics in One Complex Variable. Annals of Mathematics Studies, vol. 160, 3rd edn. Princeton University Press, Princeton (2006)zbMATHGoogle Scholar
  13. 13.
    Pommerenke, C.: Boundary Behaviour of Conformal Maps. Springer, Berlin (1992)CrossRefzbMATHGoogle Scholar
  14. 14.
    Pólya, G., Szegö, G.: Problems and Theorems in Analysis I. Springer, New York (1972)CrossRefzbMATHGoogle Scholar
  15. 15.
    Rempe-Gillen, L.: Hyperbolic entire functions with full hyperbolic dimension and approximation by Eremenko–Lyubich functions. Proc. Lond. Math. Soc. 108(5), 1193–1225 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Rottenfusser, G., Rückert, J., Rempe, L., Schleicher, D.: Dynamic rays of entire functions. Ann. Math. (2) 173(1), 77–125 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Steinmetz, N.: Rational Iteration: Complex Analytic Dynamical Systems. de Gruyter Studies in Mathematics A, vol. 16. de Gruyter, Berlin (1993)CrossRefzbMATHGoogle Scholar
  18. 18.
    Viana da Silva, M.: The differentiability of the hairs of exp(Z). Proc. Am. Math. Soc. 103(4), 1179–1184 (1988)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Weierstraß, K.: Über continuirliche Functionen eines reellen Arguments, die für keinen Werth des letzteren einen bestimmten Differentialquotienten besitzen. Math. Werke, vol. 2, pp. 71–74, Berlin (1895)Google Scholar

Copyright information

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

Authors and Affiliations

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

Personalised recommendations