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HAUSDORFF DIMENSION AND HAUSDORFF MEASURES OF JULIA SETS OF ELLIPTIC FUNCTIONS

Published online by Cambridge University Press:  20 March 2003

JANINA KOTUS  and
MARIUSZ URBAŃSKI
JANINA KOTUS
Affiliation:
Faculty of Mathematics and Information Sciences, Warsaw University of Technology, Warsaw 00-661, [email protected]
MARIUSZ URBAŃSKI
Affiliation:
Department of Mathematics, University of North Texas, P.O. Box 311430, Denton TX 76203-1430, [email protected]
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Abstract

It is proved here that if $f : {\rm C\!\!\!I} \rightarrow \bar{{\rm C\!\!\!I}}$ is an elliptic function and $q$ is the maximal multiplicity of all poles of $f$, then the Hausdorff dimension of the Julia set of $f$ is greater than $2q/(q+1)$, and the Hausdorff dimension of the set of points that escape to infinity is less than or equal to $2q/(q+1)$. In particular, the area of this latter set is equal to 0.

Keywords

37F35 (primary)37F10 (secondary)30D30 (secondary)
Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 35 , Issue 2 , March 2003 , pp. 269 - 275
Copyright
© The London Mathematical Society 2003

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Footnotes

The research of the first author was supported in part by the Foundation for Polish Science, the Polish KBN Grant No. 2 P03A 009 17 and TUW Grant No. 503G 112000442200. She also wishes to thank the University of North Texas, where this research was conducted. The research of the second author was supported in part by the NSF Grant DMS 9801583.