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Conical Limit Set and Poincare Exponent for Iterations of Rational Functions

Transactions of the American Mathematical Society / v.351 no.5. 1999, pp.2081-2099

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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 Poincare exponent $\delta (f,z)=$ $\inf \{\geq 0\colon $ $\scr{P}(z,\alpha)\leq 0\}$ , where $\scr{P}(z,\alpha)\colon =$ $\underset n\rightarrow \infty \to{\lim \text{}\sup} $ \frac{1}{n} $ \log $ $\underset f^{n}(x)=x\to{\Sigma}$ $ (f^{n})^{\prime}(x) ^{-\alpha}$ . We prove that $\delta (f,z)$ and $\scr{P}(z,\alpha)$ do not depend on z, provided z is non-exceptional. $\scr{P}$ plays the role of pressure; we prove that it coincides with the Denker-Urbanski pressure if $\alpha \leq \delta (f)$ . 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.

Keyword : Primary 58F23

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