Ergodic theory for Markov fibred systems and parabolic rational maps

Aaronson, Jon; Denker, Manfred; Urbański, Mariusz

doi:10.1090/S0002-9947-1993-1107025-2

Ergodic theory for Markov fibred systems and parabolic rational maps
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by Jon Aaronson, Manfred Denker and Mariusz Urbański PDF

Trans. Amer. Math. Soc. 337 (1993), 495-548 Request permission

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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