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

DOI:
https://doi.org/10.1090/S0002-9947-99-02195-9

Published electronically:
January 26, 1999

MathSciNet review:
1615954

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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 , where

We prove that and do not depend on , provided is non-exceptional. plays the role of pressure; we prove that it coincides with the Denker-Urbanski pressure if . Various notions of ``conical limit set" are considered. They all have Hausdorff dimension equal to 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

Email:
feliksp@impan.gov.pl

DOI:
https://doi.org/10.1090/S0002-9947-99-02195-9

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