### Abstract

Let T be a rational map of degree d ³ 2 of the Riemann sphere ₵ = ₵ u{¥}. We develop the theory of equilibrium states for the class of Holder continuous functions/for which the pressure is larger than sup f. We show that there exist a unique conformal measure (reference measure) and a unique equilibrium state, which is equivalent to the conformal measure with a positive continuous density. The associated Perron-Frobenius operator acting on the space of continuous functions is almost periodic and we show that the system is exact with respect to the equilibrium measure.

Original language | English (US) |
---|---|

Pages (from-to) | 103-134 |

Number of pages | 32 |

Journal | Nonlinearity |

Volume | 4 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 1991 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)
- Applied Mathematics

## Fingerprint Dive into the research topics of 'Ergodic theory of equilibrium states for rational maps'. Together they form a unique fingerprint.

## Cite this

Denker, M., & Urbanski, M. (1991). Ergodic theory of equilibrium states for rational maps.

*Nonlinearity*,*4*(1), 103-134. https://doi.org/10.1088/0951-7715/4/1/008