Abstract
Devaney and Krych showed that for \(0<\lambda <1/e\) the Julia set of \(\lambda e^z\) consists of pairwise disjoint curves, called hairs, which connect finite points, called the endpoints of the hairs, with \(\infty \). McMullen showed that the Julia set has Hausdorff dimension 2 and Karpińska showed that the set of hairs without endpoints has Hausdorff dimension 1. We study for which gauge functions the Hausdorff measure of the set of hairs without endpoints is finite.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bergweiler, W.: Iteration of meromorphic functions. Bull. Am. Math. Soc. (N. S.) 29, 151–188 (1993)
Bergweiler, W.: Karpińska’s paradox in dimension \(3\). Duke Math. J. 154, 599–630 (2010)
Devaney, R.L.: Complex exponential dynamics. In: Broer, H.W., Hasselblatt, B., Takens, F. (eds.) Handbook of Dynamical Systems, Chapter 4, pp. 125–223. North-Holland Elsevier Ltd., Oxford (2010)
Devaney, R.L., Goldberg, L.R.: Uniformization of attracting basis for exponential maps. Duke Math. J. 55, 253–266 (1987)
Devaney, R.L., Krych, M.: Dynamics of \(\exp (z)\). Ergodic Theory Dyn. Syst. 4, 35–52 (1984)
Falconer, K.: Fractal Geometry—Mathematical Foundations and Applications. Wiley, Chichester (1990)
Karpińska, B.: Area and Hausdorff dimension of the set of accessible points of the Julia sets of \(\lambda e^z\) and \(\lambda \sin z\). Fund. Math. 159, 269–287 (1999)
Karpińska, B.: Hausdorff dimension of the hairs without endpoints for \(\lambda \exp z\). C. R. Acad. Sci. Paris Ser. I Math. 328, 1039–1044 (1999)
Karpińska, B., Urbański, M.: How points escape to infinity under exponential maps. J. Lond. Math. Soc. (2) 73, 141–156 (2006)
McMullen, C.: Area and Hausdorff dimension of Julia sets of entire functions. Trans. Am. Math. Soc. 300, 329–342 (1987)
Peter, J.: Hausdorff measure of Julia sets in the exponential family. J. Lond. Math. Soc. (2) 82, 229–255 (2010)
Przytycki, F., Urbański, M.: Fractals in the Plane: Ergodic Theory Methods. London Mathematical Society Lecture Note Series 371. Cambridge University Press, Cambridge (2010)
Rempe, L.: Topological dynamics of exponential maps on their escaping sets. Ergodic Theory Dyn. Syst. 26, 1939–1975 (2006)
Ritt, J.F.: Transcendental transcendency of certain functions of Poincaré. Math. Ann. 95, 671–682 (1925/26)
Schleicher, D.: Attracting dynamics of exponential maps. Ann. Acad. Sci. Fenn. Math. 28, 3–34 (2003)
Schleicher, D.: Dynamics of entire functions. In: “Holomorphic dynamical systems”, Lecture Notes Mathematics 1998, pp. 295–339, Springer, Berlin (2010)
Schleicher, D., Zimmer, J.: Escaping points of exponential maps. J. Lond. Math. Soc. (2) 67, 380–400 (2003)
Stallard, G.M.: The Hausdorff dimension of Julia sets of entire functions IV. J. Lond. Math. Soc. (2) 61, 471–488 (2000)
Steinmetz, N.: Rational Iteration. Walter de Gruyter, Berlin (1993)
Acknowledgments
We thank the referee for valuable comments. The second author was supported by the scholarship from China Scholarship Council (No. 201206105015), and would express her thanks for the hospitality of Mathematisches Seminar at Christian-Albrechts-Universität zu Kiel.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bergweiler, W., Wang, J. Hausdorff measure of hairs without endpoints in the exponential family. Math. Z. 281, 931–947 (2015). https://doi.org/10.1007/s00209-015-1514-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-015-1514-8