On the transfer operator for rational functions on the Riemann sphere

Let T be a rational function of degree ≥ 2 on the Riemann sphere. Our results are based on the lemma that the diameter of a connected component of T^-n(B(x, r)), centered at any point x in its Julia set J = J (T), does not exceed Lⁿr^p for some constants L ≥ 1 and p > 0. Denote script capital L sign_φ the transfer operator of a Hölder-continuous function φ on J satisfying P(T, φ) > sup_z∈Jφ(z). We study the behavior of {script capital L signⁿ_φψ : n ≥ 1} for Hölder-continuous functions ψ and show that the sequence is (uniformly) norm-bounded in the space of Hölder-continuous functions for sufficiently small exponent. As a consequence we obtain that the density of the equilibrium measure μ for φ with respect to the exp[P(T, φ) - φ]-conformal measure is Hölder-continuous. We also prove that the rate of convergence of script capital L signⁿ_φψ to this density in sup-norm is O(exp(-θ√n)). From this we deduce the corresponding decay of the correlation integral and the central limit theorem for ψ.