### Abstract

A parabolic rational map of the Riemann sphere admits a nonatomic-conformai 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 σ-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.

Original language | English (US) |
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Pages (from-to) | 495-548 |

Number of pages | 54 |

Journal | Transactions of the American Mathematical Society |

Volume | 337 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1993 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

Aaronson, J., Denker, M., & Urbański, M. (1993). Ergodic theory for markov fibred systems and parabolic rational maps.

*Transactions of the American Mathematical Society*,*337*(2), 495-548. https://doi.org/10.1090/S0002-9947-1993-1107025-2