On the transfer operator for rational functions on the Riemann sphere

Manfred Heinz Denker, Feliks Przytycki, Mariusz Urbański

Research output: Contribution to journalArticle

54 Citations (Scopus)

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

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Transfer Operator
Rational functions
Rational function
Mathematical operators
Continuous Function
Correlation Integral
Conformal Measure
Norm
Equilibrium Measure
Spaces of Continuous Functions
Julia set
Connected Components
Central limit theorem
Lemma
Deduce
Exceed
Rate of Convergence
Exponent
Decay
Denote

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this

Denker, M. H., 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.
Denker, Manfred Heinz ; Przytycki, Feliks ; Urbański, Mariusz. / On the transfer operator for rational functions on the Riemann sphere. In: Ergodic Theory and Dynamical Systems. 1996 ; Vol. 16, No. 2. pp. 255-266.
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Denker, MH, Przytycki, F & Urbański, M 1996, 'On the transfer operator for rational functions on the Riemann sphere', 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.

In: Ergodic Theory and Dynamical Systems, Vol. 16, No. 2, 01.04.1996, p. 255-266.

Research output: Contribution to journalArticle

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AU - Urbański, Mariusz

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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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