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Transactions of the American Mathematical Society

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Ergodic theory for Markov fibred systems and parabolic rational maps


Authors: Jon Aaronson, Manfred Denker and Mariusz Urbański
Journal: Trans. Amer. Math. Soc. 337 (1993), 495-548
MSC: Primary 58F11; Secondary 28D05, 58F23, 60F05
DOI: https://doi.org/10.1090/S0002-9947-1993-1107025-2
MathSciNet review: 1107025
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Abstract: A parabolic rational map of the Riemann sphere admits a non atomic $ h$-conformal measure on its Julia set where $ h = $ the Hausdorff dimension of the Julia set and satisfies $ 1/2 < h < 2$. With respect to this measure the rational map is conservative, exact and there is an equivalent $ \sigma $-finite invariant measure. Finiteness of the measure is characterised. Central limit theorems are proved in the case of a finite invariant measure and return sequences are identified in the case of an infinite one. A theory of Markov fibred systems is developed, and parabolic rational maps are considered within this framework.


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DOI: https://doi.org/10.1090/S0002-9947-1993-1107025-2
Article copyright: © Copyright 1993 American Mathematical Society

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