On the transfer operator for rational functions on the Riemann sphere.

*(English)*Zbl 0852.46024Summary: 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 \(\psi\).

##### MSC:

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) |

##### Keywords:

Riemann sphere; Julia set; transfer operator; Hölder-continuous function; equilibrium measure; decay of the correlation integral; central limit theorem
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\textit{M. Denker} et al., Ergodic Theory Dyn. Syst. 16, No. 2, 255--266 (1996; Zbl 0852.46024)

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