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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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