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On the Perron-Frobenius-Ruelle operator for rational maps on the Riemann sphere and for Hölder continuous functions. (English) Zbl 0723.58030
Let \(f\) be a rational mapping of the Riemann sphere \({\hat {\mathbb{C}}}\) and \({\mathbb{J}}={\mathbb{J}}(f)\) be its Julia set. Let \(\phi\) be a real continuous function on \({\mathbb{J}}\). Define the operator \({\mathcal P}_{\phi}: C({\mathbb{J}})\to C({\mathbb{J}})\) on the space of continuous functions by \[ {\mathcal P}_{\phi}(\psi)(x) = \sum_{y\in f^{-1}(x)}\psi(y)\exp \phi(y) \] (if \(y\) is a critical point we repeat it as many times as its multiplicity as the preimage of \(x\)).
The author gives a new proof of the following theorem first proved by M. Denker and M. Urbański [Nonlinearity 4, No.1, 103-134 (1991; Zbl 0718.58035)]].
Theorem. Suppose sup \(\phi < P\equiv P(f,\phi)\) (the topological pressure) and suppose that \(\phi\) is Hölder continuous. Then \({\mathcal P}_{\phi -P}\) is almost periodic, i.e. for every \(\psi\in C({\mathbb{J}})\) the sequence of functions \({\mathcal P}^ n_{\phi-P}(\psi)\) is uniformly bounded and equicontinuous. There exists a positive fixed point for \({\mathcal P}_{\phi-P}\) namely a function \(\psi_ 0>0\) such that \({\mathcal P}_{\phi -P}(\psi_ 0)=\psi_ 0\) and there exists a probability measure \(\eta\) on \({\mathbb{J}}\) such that for every \(\psi\in C({\mathbb{J}})\) we have \[ \int \psi d\eta =\int {\mathcal P}_{\phi -P}(\psi)d\eta \text{ and } {\mathcal P}^ n_{\phi -P}(\psi)\to \psi_ 0\cdot \int \psi\;d\eta \text{ as } n\to \infty. \] The author’s proof uses techniques from holomorphic dynamics. The author also studies the modulus of continuity of \(\psi_ 0\), and in particular he proves that it is bounded by \(C(N)(1/(\log (1/\epsilon)))^ N\) for arbitrary \(N\) and a respective constant \(C(N)\).

MSC:
37A30 Ergodic theorems, spectral theory, Markov operators
58D15 Manifolds of mappings
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
28D20 Entropy and other invariants
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
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