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On the transfer operator for rational functions on the Riemann sphere

  • Manfred Denker (a1), Feliks Przytycki (a2) and Mariusz Urbański (a2) (a3)
  • DOI: http://dx.doi.org/10.1017/S0143385700008804
  • Published online: 01 September 2008
Abstract
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.

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[1]R. Bowen . Equilibrium states and the ergodic theory of Anosov diffeomorphsms. Springer Lecture Notes in Mathematics 470. Springer, Berlin, 1975.

[2]M. Denker , C. Grillenberger and K. Sigmund . Ergodic theory on compact spaces. Springer Lecture Notes in Mathematics 527. Springer, Berlin, 1976.

[3]M. Denker and M. Urbański . Ergodic theory of equilibrium states for rational maps. Nonlinearity 4 (1991), 103134.

[4]M. Denker and M. Urbański . The dichotomy of Hausdorff measures and equilibrium states for parabolic rational maps. Ergodic Theory and Related Topics III, Proceedings Güstrow 1990, eds. U. Krengel , K. Richter and V. Warstat . Lecture Notes in Mathematics 1514. Springer, Berlin, 1992, pp. 90113.

[8]F. Przytycki . On the Perron—Frobenius-Ruelle operator for rational maps on the Riemann sphere and for Hölder continuous functions. Bol. Soc. Bras. Mat. 20 (1990), 95125.

[9]F. Przytycki . Lyapunov characteristic exponents are non-negative. Proc. Amer. Math. Soc. 119 (1993), 309317.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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