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Ergodic theory for Markov fibred systems and parabolic rational maps. (English) Zbl 0789.28010
A rational map $$T$$ of the Riemann sphere $$\overline{\mathbb{C}}$$ of degree $$\geq 2$$ is called parabolic if the restriction to the Julia set $$J(T)$$ is expansive but not expanding in the spherical metric on $$\overline\mathbb{C}$$. For $$t\geq 0$$, a probability measure $$m$$ on $$J(T)$$ is called $$t$$-conformal for $$T$$ if $$m(T(A))=\int_ A | T'|^ t dm$$ for Borel $$A\subset J(T)$$ provided the restriction of $$T$$ to $$A$$ is injective. If $$h$$ is the Hausdorff dimension of $$J(T)$$, the unique $$h$$-conformal measure $$m$$ is shown to be nonatomic. It is known that, for this $$m$$, there is a topological Markov partition with respect to which $$(J(T),m,T)$$ is a fibred system. $$T$$ has a $$\sigma$$-finite invariant measure $$\mu\sim m$$. $$T$$ is shown to be conservative and exact and the finiteness of $$\mu$$ is characterized. For example, if $$T$$ is a parabolic Blaschke product, $$\mu$$ is infinite. For polynomials $$z\to z+ z^ 2$$, $$z\to z- z^ 2$$, or $$z\to z^ 2+ 1/4$$, $$\mu$$ is finite.
If $$\mu$$ is finite, a central limit theorem is shown for partial sums of Hölder continuous functions vanishing on neighborhoods of the rationally indifferent periodic points. If $$\mu$$ is infinite, then $$T$$ has Darling Kac sets with continued fraction mixing return time processes and return sequences of the form $$\{n^{\alpha-1}\}$$ for $$1< \alpha< 2$$ and $$\{n/\log n\}$$ for $$\alpha= 2$$. ($$\alpha$$ is given in terms of a Taylor expansion.)
A theory of Markov fibred systems is developed and parabolic rational maps are studied within this framework, extending and using results of F. Schweiger [Isr. J. Math. 21, 308-318 (1975; Zbl 0314.10037)]. There are numerous additional results.

##### MSC:
 28D05 Measure-preserving transformations 30C99 Geometric function theory 37A99 Ergodic theory 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 60F05 Central limit and other weak theorems
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