Skip to main content

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
Cancel
×
×
Home
  • Home
Article
  • Cited by 1
  • Cited by
    Crossref Citations
    Crossref logo
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.
    Bishop, Christopher J. 2018. A transcendental Julia set of dimension 1. Inventiones mathematicae, Vol. 212, Issue. 2, p. 407.
    ×
  • Access

ON THE UNIFORM PERFECTNESS OF THE BOUNDARY OF MULTIPLY CONNECTED WANDERING DOMAINS

  • WALTER BERGWEILER (a1) and JIAN-HUA ZHENG (a2)
Abstract

We investigate when the boundary of a multiply connected wandering domain of an entire function is uniformly perfect. We give a general criterion implying that it is not uniformly perfect. This criterion applies in particular to examples of multiply connected wandering domains given by Baker. We also provide examples of infinitely connected wandering domains whose boundary is uniformly perfect.

    •
      • 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 UNIFORM PERFECTNESS OF THE BOUNDARY OF MULTIPLY CONNECTED WANDERING DOMAINS
      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 UNIFORM PERFECTNESS OF THE BOUNDARY OF MULTIPLY CONNECTED WANDERING DOMAINS
      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 UNIFORM PERFECTNESS OF THE BOUNDARY OF MULTIPLY CONNECTED WANDERING DOMAINS
      Available formats
      ×
Copyright
COPYRIGHT: © Australian Mathematical Publishing Association Inc. 2011
Corresponding author
✉For correspondence; e-mail: bergweiler@math.uni-kiel.de
Footnotes
Hide All

The first-named author was supported by a Chinese Academy of Sciences Visiting Professorship for Senior International Scientists, Grant No. 2010 TIJ10, the Deutsche Forschungsgemeinschaft, Be 1508/7-1, the EU Research Training Network CODY and the ESF Networking Programme HCAA. The second-named author was supported by Grant No. 10871108 of the NSF of China.

Footnotes
References
Hide All
[1]Baker, I. N., ‘Multiply connected domains of normality in iteration theory’, Math. Z. 81 (1963), 206–214.
[2]Baker, I. N., ‘The domains of normality of an entire function’, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), 277–283.
[3]Baker, I. N., ‘An entire function which has wandering domains’, J. Aust. Math. Soc. (Ser. A) 22 (1976), 173–176.
[4]Baker, I. N., ‘Wandering domains in the iteration of entire functions’, Proc. Lond. Math. Soc. (3) 49 (1984), 563–576.
[5]Baker, I. N., ‘Infinite limits in the iteration of entire functions’, Ergodic Theory Dynam. Systems 8 (1988), 503–507.
[6]Beardon, A. F. and Pommerenke, Ch., ‘The Poincaré metric of plane domains’, J. Lond. Math. Soc. 18 (1978), 475–483.
[7]Bergweiler, W., ‘Iteration of meromorphic functions’, Bull. Amer. Math. Soc. (N.S.) 29 (1993), 151–188.
[8]Bergweiler, W., ‘Connectivity of Fatou components’, Oberwolfach Rep. 6 (2009), 2946–2948.
[9]Bergweiler, W., Rippon, P. J. and Stallard, G. M., ‘Multiply connected wandering domains of entire functions’, Preprint, arXiv:1109.1794.
[10]Eremenko, A., unpublished manuscript, http://www.math.purdue.edu/∼eremenko/dvi/ups.pdf.
[11]Hinkkanen, A., ‘Julia sets of rational functions are uniformly perfect’, Math. Proc. Cambridge Philos. Soc. 113 (1993), 543–559.
[12]Kisaka, M. and Shishikura, M., ‘On multiply connected wandering domains of entire functions’, in: Transcendental Dynamics and Complex Analysis, LMS Lecture Note Series, 348 (eds. Rippon, P. J. and Stallard, G. M.) (Cambridge University Press, Cambridge, 2008), pp. 217–250.
[13]Mañé, R. and da Rocha, L. F., ‘Julia sets are uniformly perfect’, Proc. Amer. Math. Soc. 116 (1992), 251–257.
[14]McMullen, C. T., Complex Dynamics and Renormalization, Annals of Mathematics Studies, 135 (Princeton University Press, Princeton, NJ, 1994).
[15]Pommerenke, Ch., ‘Uniformly perfect sets and the Poincaré metric’, Arch. Math. (Basel) 32 (1979), 192–199.
[16]Schleicher, D., ‘Dynamics of entire functions’, in: Holomorphic Dynamical Systems, Lecture Notes in Mathematics, 1998 (Springer, Heidelberg–Dordrecht–London–New York, 2010), pp. 295–339.
[17]Steinmetz, N., Rational Iteration (Walter de Gruyter, Berlin, 1993).
[18]Sugawa, T., ‘Various domain constants related to uniform perfectness’, Complex Variables Theory Appl. 36 (1998), 311–345.
[19]Sullivan, D., ‘Quasiconformal homeomorphisms and dynamics I. Solution of the Fatou–Julia problem on wandering domains’, Ann. of Math. (2) 122 (1985), 401–418.
[20]Töpfer, H., ‘Über die Iteration der ganzen transzendenten Funktionen, insbesondere von sin z und cos z’, Math. Ann. 117 (1939), 65–84.
[21]Zheng, J. H., ‘On uniformly perfect boundaries of stable domains in iteration of meromorphic functions’, Bull. Lond. Math. Soc. 32 (2000), 439–446.
[22]Zheng, J. H., ‘Uniformly perfect sets and distortion of holomorphic functions’, Nagoya Math. J. 164 (2001), 17–33.
[23]Zheng, J. H., ‘On multiply-connected Fatou components in iteration of meromorphic functions’, J. Math. Anal. Appl. 313 (2006), 24–37.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

CAPTCHA *

Skip to the audio challenge
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 38 *
Loading metrics...

Abstract views

Total abstract views: 66 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 13th June 2018. This data will be updated every 24 hours.

Cancel
Confirm
×