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On the transfer operator for rational functions on the Riemann sphere. (English) Zbl 0852.46024
Summary: Let \(T\) be a rational function of degree \(\geq 2\) 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 \(L^* r^\rho\) for some constants \(L\geq 1\) and \(\rho> 0\). Denote \({\mathcal L}_\phi\) the transfer operator of a Hölder-continuous function \(\phi\) on \(J\) satisfying \(P(T, \phi)> \sup_{z\in J} \phi(z)\). We study the behavior of \(\{{\mathcal L}^n_\phi \psi: n\geq 1\}\) for Hölder-continuous functions \(\psi\) 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 for the equilibrium measure \(\mu\) for \(\phi\) with respect to the \(\exp[P(T, \phi)- \phi]\)-conformal measure is Hölder-continuous. We also prove that the rate of convergence of \({\mathcal L}^n_\phi \psi\) to this density in sup-norm is \(O(\exp(- \theta \sqrt n))\). From this, we deduce the corresponding decay of the correlation integral and the central limit theorem for \(\psi\).

46E10 Topological linear spaces of continuous, differentiable or analytic functions
60F05 Central limit and other weak theorems
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
Full Text: DOI
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