### Abstract

Iteration of a rational function R gives a complex dynamical system on the Riemann sphere. We introduce a C*-algebra script O sign_{R} associated with R as a Cuntz-Pimsner algebra of a Hubert bimodule over the algebra A = C(J_{R}) of continuous functions on the Julia set J _{R} of R. The algebra O_{R} is a certain analog of the crossed product by a boundary action. We show that if the degree of R is at least two, then C* -algebra OR is simple and purely infinite. For example if R(z) = z^{2} - 2, then the Julia set J_{R} = [-2,2] and the restriction R: J_{R} → J_{R} is topologically conjugate to the tent map on [0,1]. The algebra O_{Z2-2} is isomorphic to the Cuntz algebra O∞. We also show that the Lyubich measure associated with R gives a unique KMS state on the C*-algebra O_{R} for the gauge action at inverse temperature log(deg-R), if the Julia set contains no critical points. Indiana University Mathematics Journal

Original language | English |
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Pages (from-to) | 755-778 |

Number of pages | 24 |

Journal | Indiana University Mathematics Journal |

Volume | 54 |

Issue number | 3 |

DOIs | |

Publication status | Published - Aug 16 2005 |

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### Keywords

- C*-algebras
- Complex dynamical systems
- Cuntz-Pimsner algebras
- Rational functions

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Indiana University Mathematics Journal*,

*54*(3), 755-778. https://doi.org/10.1512/iumj.2005.54.2530