Ergodic Theory and Dynamical Systems

Research Article

Rational functions with no recurrent critical points

Mariusz Urbańskia1

a1 Department of Mathematics, University of North Texas, Denton, TX 76203-5116, USA

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.

(Received November 17 1992)

Footnotes

† Research supported by NSF grant DMS 9303888 and KBN grant 211109101.