Skip to main content Accessibility help
×
Home
Hostname: page-component-564cf476b6-2jsqd Total loading time: 0.168 Render date: 2021-06-20T14:24:24.683Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

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
Affiliation:
Mathematisches Seminar, Christian–Albrechts–Universität zu Kiel, Ludewig–Meyn–Str. 4, D–24098 Kiel, Germany. e-mail: bergweiler@math.uni-kiel.de
BOGUSŁAWA KARPIŃSKA
Affiliation:
Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warszawa, Poland. e-mail: bkarpin@mini.pw.edu.pl
Corresponding

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
Copyright
Copyright © Cambridge Philosophical Society 2010

Access options

Get access to the full version of this content by using one of the access options below.

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), 719737.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), 615624.Google Scholar
[4]Bergweiler, W.Iteration of meromorphic functions. Bull. Am. Math. Soc. (N. S.) 29 (1993), 151188.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), 680698.CrossRefGoogle Scholar
[6]Eremenko, A. E. and Lyubich, M. Yu.Dynamical properties of some classes of entire functions. Ann. Inst. Fourier 42 (1992), 9891020.CrossRefGoogle Scholar
[7]Fatou, P.Sur l'itération des fonctions transcendantes entières. Acta Math. 47 (1926), 337360.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), 162168.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), 93128.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), 329342.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), 195214.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), 513536.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. 425446.CrossRefGoogle Scholar
[20]Taniguchi, M.Size of the Julia set of structurally finite transcendental entire function. Math. Proc. Camb. Phil. Soc. 135 (2003), 181192.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), 347361.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), 2437.CrossRefGoogle Scholar
11
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

On the Hausdorff dimension of the Julia set of a regularly growing entire function
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

On the Hausdorff dimension of the Julia set of a regularly growing entire function
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

On the Hausdorff dimension of the Julia set of a regularly growing entire function
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *