### Abstract

We consider an analogy among Markov shifts, complex dynamical systems and self-similar maps. Their dynamics are given by 0–1 matrices A, rational functions R and self-similar maps γ on a compact metric space K, respectively. If the 0–1 matrix A is irreducible and not a permutation, then the Cuntz–Krieger algebra O_{A} is simple and purely infinite. Similarly, if the rational function R is restricted to the Julia set J_{R} and the self-similar map γ satisfies the open set condition respectively, then the associated C^{⁎}-algebras O_{R}(J_{R}) and O_{γ}(K) are simple and purely infinite. Let Σ_{A} be the associated infinite path space for the 0–1 matrix A, then C(Σ_{A}) is known to be a maximal abelian subalgebra of O_{A}. In this paper we shall show that C(J_{R}) is a maximal abelian subalgebra of O_{R}(J_{R}) and C(K) is a maximal abelian subalgebra of O_{γ}(K).

Original language | English |
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Pages (from-to) | 1383-1400 |

Number of pages | 18 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 455 |

Issue number | 2 |

DOIs | |

Publication status | Published - Nov 15 2017 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

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## Cite this

^{⁎}-algebras associated with complex dynamical systems and self-similar maps.

*Journal of Mathematical Analysis and Applications*,

*455*(2), 1383-1400. https://doi.org/10.1016/j.jmaa.2017.06.044