Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 58F23

Retrieve articles in all journals with MSC (1991): 58F23


Additional Information

Feliks Przytycki
Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00950 Warsaw, Poland
Email: feliksp@impan.gov.pl

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02195-9
PII: S 0002-9947(99)02195-9
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