On the transfer operator for rational functions on the Riemann sphere

Manfred Denker, Feliks Przytycki, Mariusz Urbański

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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 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.

Original languageEnglish (US)
Pages (from-to)255-266
Number of pages12
JournalErgodic Theory and Dynamical Systems
Volume16
Issue number2
StatePublished - Apr 1 1996

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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    Denker, M., Przytycki, F., & Urbański, M. (1996). On the transfer operator for rational functions on the Riemann sphere. Ergodic Theory and Dynamical Systems, 16(2), 255-266.