Mathematical Proceedings of the Cambridge Philosophical Society



The geometry of conformal measures for parabolic rational maps


B. O. STRATMANN a1 and M. URBANSKI a2 1
a1 Mathematical Institute, University of St Andrews, St Andrews KY16 9SS, Scotland; e-mail: bos@maths.st-and.ac.uk
a2 Department of Mathematics, University of North Texas, Denton, TX 76203-5118, U.S.A. e-mail: urbanski@unt.edu

Abstract

We study the h-conformal measure for parabolic rational maps, where h denotes the Hausdorff dimension of the associated Julia sets. We derive a formula which describes in a uniform way the scaling of this measure at arbitrary elements of the Julia set. Furthermore, we establish the Khintchine Limit Law for parabolic rational maps (the analogue of the ‘logarithmic law for geodesics’ in the theory of Kleinian groups) and show that this law provides some efficient control for the fluctuation of the h-conformal measure. We then show that these results lead to some refinements of the description of this measure in terms of Hausdorff and packing measures with respect to some gauge functions. Also, we derive a simple proof of the fact that the Julia set of a parabolic rational map is uniformly perfect. Finally, we obtain that the conformal measure is a regular doubling measure, we show that its Renyi dimension and its information dimension are equal to h and we compute its logarithmic index.

(Received July 28 1998)
(Revised October 20 1998)



Footnotes

1 Research partially supported by the NSF Grant DMS 9801583.