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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Conical limit set and Poincaré exponent
for iterations of rational functions

Author: Feliks Przytycki
Journal: Trans. Amer. Math. Soc. 351 (1999), 2081-2099
MSC (1991): Primary 58F23
Published electronically: January 26, 1999
MathSciNet review: 1615954
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Abstract: We contribute to the dictionary between action of Kleinian groups and iteration of rational functions on the Riemann sphere. We define the Poincaré exponent $\delta(f,z)=\inf\{\alpha\ge 0:\mathcal{P}(z,\alpha) \le 0\}$, where

\begin{equation*}\mathcal{P}(z,\alpha):=\limsup _{n\to\infty}{1\over n}\log\sum _{f^n(x)=z} |(f^n)'(x)|^{-\alpha}. \end{equation*}

We prove that $\delta (f,z)$ and $\mathcal{P}(z,\alpha)$ do not depend on $z$, provided $z$ is non-exceptional. $\mathcal{P}$ plays the role of pressure; we prove that it coincides with the Denker-Urbanski pressure if $\alpha\le \delta(f)$. Various notions of ``conical limit set" are considered. They all have Hausdorff dimension equal to $\delta(f)$ which is equal to the hyperbolic dimension of the Julia set and also equal to the exponent of some conformal Patterson-Sullivan measures. In an Appendix we also discuss notions of ``conical limit set" introduced recently by Urbanski and by Lyubich and Minsky.

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Additional Information

Feliks Przytycki
Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00950 Warsaw, Poland

Received by editor(s): December 2, 1996
Published electronically: January 26, 1999
Additional Notes: Supported by Polish KBN Grant 2 P301 01307 and by the Max-Planck-Institut für Mathematik in Bonn, where the author stayed in Summer 1996
Article copyright: © Copyright 1999 American Mathematical Society