### Abstract

Let R be a rational function. The iterations (R^{n})_{n} of R gives a complex dynamical system on the Riemann sphere. We associate a C^{*}-algebra and study a relation between the C^{*}-algebra and the original complex dynamical system. In this short note, we recover the number of n th backward orbits counted without multiplicity starting at branched points in terms of associated C^{*}-algebras with gauge actions. In particular, we can partially imagine how a branched point is moved to another branched point under the iteration of R. We use KMS states and a Perron-Frobenius type operator on the space of traces to show it.

Original language | English |
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Pages (from-to) | 243-254 |

Number of pages | 12 |

Journal | Complex Analysis and Operator Theory |

Volume | 8 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2014 |

### Keywords

- Backward orbit
- Branched point
- C-algebra
- C-correspondences
- Complex dynamical system

### ASJC Scopus subject areas

- Computational Mathematics
- Computational Theory and Mathematics
- Applied Mathematics

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## Cite this

Kajiwara, T., & Watatani, Y. (2014). C

^{*}-Algebras Associated with Complex Dynamical Systems and Backward Orbit Structure.*Complex Analysis and Operator Theory*,*8*(1), 243-254. https://doi.org/10.1007/s11785-013-0293-7