Ergodic Theory and Dynamical Systems

Research Article

On the transfer operator for rational functions on the Riemann sphere

Manfred Denkera1, Feliks Przytyckia2 and Mariusz Urbańskia2a3

a1 Institut für Mathematische Stochastik, Universität Göttingen, Lotzestraβe 13, 37083 Göttingen, Germany

a2 Institute of Mathematics, Polish Academy of Science, ul. Śniadeckich 8, 00-950 Warsaw, Poland

a3 Department of Mathematics, University of North Texas, Denton TX 76203-5116, USA

Abstract

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 Lnrp for some constants L ≥ 1 and ρ > 0. Denote the transfer operator of a Hölder-continuous function φ on J satisfying P(T,φ) > supzJφ(z). We study the behavior of {: n ≥ 1} for Hölder-continuous functions ψ and show that the sequence is (uniformly) normbounded 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 -conformal measure is Höolder-continuous. We also prove that the rate of convergence of {ψ to this density in sup-norm is . From this we deduce the corresponding decay of the correlation integral and the central limit theorem for ψ.

(Received April 07 1994)

(Revised October 04 1994)

Footnotes

† Research supported by Polish KBN Grant 210469101 ‘Iteracje i Fraktale’ and by Deutsche Forschungsgemeinschaft, SFB 170.

‡ Research supported by NSF Grant DMS 9303888 and ONR-N00014-93-1-0707.