### 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 L^{n}r^{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∈J}φ(z). We study the behavior of {script capital L sign^{n}_{φ}ψ : 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 sign^{n}_{φ}ψ 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.

Original language | English (US) |
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Pages (from-to) | 255-266 |

Number of pages | 12 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 16 |

Issue number | 2 |

State | Published - Apr 1 1996 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Ergodic Theory and Dynamical Systems*,

*16*(2), 255-266.