### 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 2014 |

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### Keywords

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

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics
- Computational Theory and Mathematics

### Cite this

*Complex Analysis and Operator Theory*,

*8*(1), 243-254. https://doi.org/10.1007/s11785-013-0293-7

**C*-Algebras Associated with Complex Dynamical Systems and Backward Orbit Structure.** / Kajiwara, Tsuyoshi; Watatani, Yasuo.

Research output: Contribution to journal › Article

*Complex Analysis and Operator Theory*, vol. 8, no. 1, pp. 243-254. https://doi.org/10.1007/s11785-013-0293-7

}

TY - JOUR

T1 - C*-Algebras Associated with Complex Dynamical Systems and Backward Orbit Structure

AU - Kajiwara, Tsuyoshi

AU - Watatani, Yasuo

PY - 2014/1

Y1 - 2014/1

N2 - Let R be a rational function. The iterations (Rn)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.

AB - Let R be a rational function. The iterations (Rn)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.

KW - Backward orbit

KW - Branched point

KW - C-algebra

KW - C-correspondences

KW - Complex dynamical system

UR - http://www.scopus.com/inward/record.url?scp=84891631480&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84891631480&partnerID=8YFLogxK

U2 - 10.1007/s11785-013-0293-7

DO - 10.1007/s11785-013-0293-7

M3 - Article

AN - SCOPUS:84891631480

VL - 8

SP - 243

EP - 254

JO - Complex Analysis and Operator Theory

JF - Complex Analysis and Operator Theory

SN - 1661-8254

IS - 1

ER -