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Rational functions with no recurrent critical points

Published online by Cambridge University Press:  19 September 2008

Mariusz Urbański
Mariusz Urbański
Affiliation:
Department of Mathematics, University of North Texas, Denton, TX 76203-5116, USA
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Abstract

Let h be the Hausdorff dimension of the Julia set of a rational map with no nonperiodic recurrent critical points. We give necessary and sufficient conditions for h-dimensional Hausdorff measure and h-dimensional packing measure of the Julia set to be positive and finite. We also show that either the Julia set is the whole Riemann sphere or h < 2 and that if a rational map (not necessarily with no nonperiodic recurrent critical points!) has a rationally indifferent periodic point, then h > 1/2.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 2 , June 1994 , pp. 391 - 414
Copyright
Copyright © Cambridge University Press 1994

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