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.
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References
Barański, K.: Trees and hairs for some hyperbolic entire maps of finite order. Math. Z. 257(1), 33–59 (2007)
Beardon, A.F.: Iteration of Rational Functions: Complex Analytic Dynamical Systems. Graduate Texts in Mathematics, vol. 132. Springer, New York (1991)
Bergweiler, W.: Iteration of meromorphic functions. Bull. Am. Math. Soc. (N. S.) 29, 151–188 (1993)
Barański, K., Jarque, X., Rempe, L.: Brushing the hairs of transcendental entire functions. Topol. Appl. 159(8), 2102–2114 (2012)
Comdühr, P.: On the differentiability of hairs for Zorich maps. Ergod. Theory Dyn. Syst. (2017). https://doi.org/10.1017/etds.2017.104
Devaney, R.L., Krych, M.: Dynamics of exp(z). Ergod. Theory Dyn. Syst. 4, 35–52 (1984)
Duren, P.: Harmonic Mappings in the Plane. Cambridge University Press, Cambridge (2004)
Eremenko, A.È., Lyubich, M.Y.: Dynamical properties of some classes of entire functions. Ann. Inst. Fourier (Grenoble) 42, 989–1020 (1992)
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)
Goldberg, A.A., Ostrovskii, I.V.: Value Distribution of Meromorphic Functions. Translations of Mathematical Monographs, vol. 236. American Mathematical Society, Providence (2008)
Mihaljević-Brandt, H.: Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds. Trans. Am. Math. Soc. 364(8), 4053–4083 (2012)
Milnor, J.: Dynamics in One Complex Variable. Annals of Mathematics Studies, vol. 160, 3rd edn. Princeton University Press, Princeton (2006)
Pommerenke, C.: Boundary Behaviour of Conformal Maps. Springer, Berlin (1992)
Pólya, G., Szegö, G.: Problems and Theorems in Analysis I. Springer, New York (1972)
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)
Rottenfusser, G., Rückert, J., Rempe, L., Schleicher, D.: Dynamic rays of entire functions. Ann. Math. (2) 173(1), 77–125 (2011)
Steinmetz, N.: Rational Iteration: Complex Analytic Dynamical Systems. de Gruyter Studies in Mathematics A, vol. 16. de Gruyter, Berlin (1993)
Viana da Silva, M.: The differentiability of the hairs of exp(Z). Proc. Am. Math. Soc. 103(4), 1179–1184 (1988)
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)
Acknowledgements
I would like to thank Walter Bergweiler, Lasse Rempe-Gillen, Dan Nicks and the referee for valuable suggestions.
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Comdühr, P. Nowhere differentiable hairs for entire maps. Math. Z. 292, 343–359 (2019). https://doi.org/10.1007/s00209-018-2223-x
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DOI: https://doi.org/10.1007/s00209-018-2223-x