Ergodic theory for Markov fibred systems and parabolic rational maps

Abstract

A parabolic rational map of the Riemann sphere admits a non atomic h h -conformal measure on its Julia set where h = h = the Hausdorff dimension of the Julia set and satisfies 1 / 2 > h > 2 1/2 > h > 2 . With respect to this measure the rational map is conservative, exact and there is an equivalent σ \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.