Conical limit set and Poincaré exponent for iterations of rational functions

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 δ ( f , z ) = inf { α ≥ 0 : P ( z , α ) ≤ 0 } \delta (f,z)=\inf \{ \alpha \ge 0: \mathcal {P}(z, \alpha ) \le 0\} , where P ( z , α ) := lim sup n → ∞ 1 n log ⁡ ∑ f n ( x ) = z | ( f n ) ′ ( x ) | − α . \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 δ ( f , z ) \delta (f,z) and P ( z , α ) \mathcal {P}(z, \alpha ) do not depend on z z , provided z z is non-exceptional. P \mathcal {P} plays the role of pressure; we prove that it coincides with the Denker-Urbański pressure if α ≤ δ ( f ) \alpha \le \delta (f) . Various notions of “conical limit set" are considered. They all have Hausdorff dimension equal to δ ( f ) \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.