Conical limit set and Poincaré exponent for iterations of rational functions

Przytycki, Feliks

doi:10.1090/S0002-9947-99-02195-9

Conical limit set and Poincaré exponent for iterations of rational functions
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Trans. Amer. Math. Soc. 351 (1999), 2081-2099 Request permission

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-Urbański 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 Urbański and by Lyubich and Minsky.

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