### 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 |
---|---|

Pages (from-to) | 755-778 |

Number of pages | 24 |

Journal | Indiana University Mathematics Journal |

Volume | 54 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2005 |

### Fingerprint

### 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

**C* -algebras associated with complex dynamical systems.** / Kajiwara, Tsuyoshi; Watatani, Yasuo.

Research output: Contribution to journal › Article

*Indiana University Mathematics Journal*, vol. 54, no. 3, pp. 755-778. https://doi.org/10.1512/iumj.2005.54.2530

}

TY - JOUR

T1 - C* -algebras associated with complex dynamical systems

AU - Kajiwara, Tsuyoshi

AU - Watatani, Yasuo

PY - 2005

Y1 - 2005

N2 - Iteration of a rational function R gives a complex dynamical system on the Riemann sphere. We introduce a C*-algebra script O signR associated with R as a Cuntz-Pimsner algebra of a Hubert bimodule over the algebra A = C(JR) of continuous functions on the Julia set J R of R. The algebra OR 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) = z2 - 2, then the Julia set JR = [-2,2] and the restriction R: JR → JR is topologically conjugate to the tent map on [0,1]. The algebra OZ2-2 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 OR for the gauge action at inverse temperature log(deg-R), if the Julia set contains no critical points. Indiana University Mathematics Journal

AB - Iteration of a rational function R gives a complex dynamical system on the Riemann sphere. We introduce a C*-algebra script O signR associated with R as a Cuntz-Pimsner algebra of a Hubert bimodule over the algebra A = C(JR) of continuous functions on the Julia set J R of R. The algebra OR 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) = z2 - 2, then the Julia set JR = [-2,2] and the restriction R: JR → JR is topologically conjugate to the tent map on [0,1]. The algebra OZ2-2 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 OR for the gauge action at inverse temperature log(deg-R), if the Julia set contains no critical points. Indiana University Mathematics Journal

KW - C-algebras

KW - Complex dynamical systems

KW - Cuntz-Pimsner algebras

KW - Rational functions

UR - http://www.scopus.com/inward/record.url?scp=23244457914&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23244457914&partnerID=8YFLogxK

U2 - 10.1512/iumj.2005.54.2530

DO - 10.1512/iumj.2005.54.2530

M3 - Article

AN - SCOPUS:23244457914

VL - 54

SP - 755

EP - 778

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 3

ER -