### 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) |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Ergodic Theory and Dynamical Systems*,

*16*(2), 255-266.

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*Ergodic Theory and Dynamical Systems*, vol. 16, no. 2, pp. 255-266.

**On the transfer operator for rational functions on the Riemann sphere.** / Denker, Manfred Heinz; Przytycki, Feliks; Urbański, Mariusz.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the transfer operator for rational functions on the Riemann sphere

AU - Denker, Manfred Heinz

AU - Przytycki, Feliks

AU - Urbański, Mariusz

PY - 1996/4/1

Y1 - 1996/4/1

N2 - 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 p > 0. Denote script capital L signφ the transfer operator of a Hölder-continuous function φ on J satisfying P(T, φ) > supz∈Jφ(z). We study the behavior of {script capital L signnφψ : 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 signnφψ 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.

AB - 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 p > 0. Denote script capital L signφ the transfer operator of a Hölder-continuous function φ on J satisfying P(T, φ) > supz∈Jφ(z). We study the behavior of {script capital L signnφψ : 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 signnφψ 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.

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M3 - Article

VL - 16

SP - 255

EP - 266

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 2

ER -