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On the Hausdorff dimension of the Julia set of a regularly growing entire function†

Published online by Cambridge University Press:  15 January 2010

WALTER BERGWEILER  and
BOGUSŁAWA KARPIŃSKA
WALTER BERGWEILER
Affiliation:
Mathematisches Seminar, Christian–Albrechts–Universität zu Kiel, Ludewig–Meyn–Str. 4, D–24098 Kiel, Germany. e-mail: [email protected]
BOGUSŁAWA KARPIŃSKA
Affiliation:
Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warszawa, Poland. e-mail: [email protected]
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Abstract

We show that if the growth of a transcendental entire function f is sufficiently regular, then the Julia set and the escaping set of f have Hausdorff dimension 2.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 3 , May 2010 , pp. 531 - 551
Copyright
Copyright © Cambridge Philosophical Society 2010

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References

REFERENCES

[1]Aspenberg, M. and Bergweiler, W. Entire functions with Julia sets of positive measure. Preprint, arxiv: 0904.1295.Google Scholar
[2]Barański, K.Hausdorff dimension of hairs and ends for entire maps of finite order. Math. Proc. Camb. Phil. Soc. 145 (2008), 719–737.CrossRefGoogle Scholar
[3]Barański, K., Karpińska, B. and Zdunik, A.Hyperbolic dimension of Julia sets of meromorphic maps with logarithmic tracts. Int. Math. Res. Not. 2009 (2009), 615–624.Google Scholar
[4]Bergweiler, W.Iteration of meromorphic functions. Bull. Am. Math. Soc. (N. S.) 29 (1993), 151–188.CrossRefGoogle Scholar
[5]Bergweiler, W., Karpińska, B. and Stallard, G. M.The growth rate of an entire function and the Hausdorff dimension of its Julia set. J. London Math. Soc. 80 (2009), 680–698.CrossRefGoogle Scholar
[6]Eremenko, A. E. and Lyubich, M. Yu.Dynamical properties of some classes of entire functions. Ann. Inst. Fourier 42 (1992), 989–1020.CrossRefGoogle Scholar
[7]Fatou, P.Sur l'itération des fonctions transcendantes entières. Acta Math. 47 (1926), 337–360.CrossRefGoogle Scholar
[8]Macintyre, A. J. and Fuchs, W. H. J.Inequalities for the logarithmic derivatives of a polynomial. J. London Math. Soc. 15 (1940), 162–168.CrossRefGoogle Scholar
[9]Goldberg, A. A. and Ostrovskii, I. V.Value distribution of meromorphic functions. Transl. Math. Monogr. 236, (Am. Math. Soc., 2008).CrossRefGoogle Scholar
[10]Hayman, W. K.Meromorphic Functions. (Clarendon Press, 1964).Google Scholar
[11]Hayman, W. K.On the characteristic of functions meromorphic in the plane and of their integrals. Proc. London Math. Soc. (3) 14a (1965), 93–128.CrossRefGoogle Scholar
[12]Levin, B. Ja.Distribution of zeros of entire functions (Am. Math. Soc., 1964)CrossRefGoogle Scholar
[13]McMullen, C.Area and Hausdorff dimension of Julia sets of entire functions. Trans. Am. Math. Soc. 300 (1987), 329–342.CrossRefGoogle Scholar
[14]Miles, J. and Rossi, J.Linear combinations of logarithmic derivatives of entire functions with applications to differential equations. Pacific J. Math. 174 (1996), 195–214.CrossRefGoogle Scholar
[15]Rempe, L. Rigidity of escaping dynamics for transcendental entire functions. To appear in Acta Math., arXiv: math/0605058.Google Scholar
[16]Rottenfusser, G., Rückert, J., Rempe, L. and Schleicher, D. Dynamic rays of bounded-type entire functions. To appear in Ann. of Math., arXiv: 0704.3213.Google Scholar
[17]Schubert, H.Über die Hausdorff-Dimension der Juliamenge von Funktionen endlicher Ordnung. Dissertation, University of Kiel, 2007; http://eldiss.uni-kiel.de/macau/receive/dissertation_diss_00002124.Google Scholar
[18]Stallard, G. M.The Hausdorff dimension of Julia sets of entire functions II. Math. Proc. Camb. Phil. Soc. 119 (1996), 513–536.CrossRefGoogle Scholar
[19]Stallard, G. M. Dimensions of Julia sets of transcendental meromorphic functions. In Transcendental Dynamics and Complex Analysis. London Math. Soc. Lect. Note Ser. 348. Edited by Rippon, P. J. and Stallard, G. M. (Cambridge University Press, 2008), pp. 425–446.CrossRefGoogle Scholar
[20]Taniguchi, M.Size of the Julia set of structurally finite transcendental entire function. Math. Proc. Camb. Phil. Soc. 135 (2003), 181–192.CrossRefGoogle Scholar
[21]Townsend, D.Comparisons between T(r, f) and the total variation of arg f(re iθ) and log |f(re iθ)|. J. Math. Anal. Appl. 128 (1987), 347–361.CrossRefGoogle Scholar
[22]Valiron, G.Lectures on the general theory of integral functions. Édouard Privat, Toulouse, 1923; (Chelsea, 1949).Google Scholar
[23]Zheng, J.-H.On multiply-connected Fatou components in iteration of meromorphic functions. J. Math. Anal. Appl. 313 (2006), 24–37.CrossRefGoogle Scholar