### 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 OA is simple and purely infinite. Similarly, if the rational function R is restricted to the Julia set JR and the self-similar map γ satisfies the open set condition respectively, then the associated C*-algebras OR(JR) and Oγ(K) are simple and purely infinite. Let ΣA be the associated infinite path space for the 0-1 matrix A, then C(ΣA) is known to be a maximal abelian subalgebra of OA. In this paper we shall show that C(JR) is a maximal abelian subalgebra of OR(JR) and C(K) is a maximal abelian subalgebra of Oγ(K).

Original language | English |
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Journal | Journal of Mathematical Analysis and Applications |

DOIs | |

Publication status | Accepted/In press - 2017 |

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### Keywords

- Complex dynamical systems
- Maximal abelian subalgebras
- Self-similar maps

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**Maximal abelian subalgebras of C ^{*}-algebras associated with complex dynamical systems and self-similar maps.** / Kajiwara, Tsuyoshi; Watatani, Yasuo.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Maximal abelian subalgebras of C*-algebras associated with complex dynamical systems and self-similar maps

AU - Kajiwara, Tsuyoshi

AU - Watatani, Yasuo

PY - 2017

Y1 - 2017

N2 - 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 OA is simple and purely infinite. Similarly, if the rational function R is restricted to the Julia set JR and the self-similar map γ satisfies the open set condition respectively, then the associated C*-algebras OR(JR) and Oγ(K) are simple and purely infinite. Let ΣA be the associated infinite path space for the 0-1 matrix A, then C(ΣA) is known to be a maximal abelian subalgebra of OA. In this paper we shall show that C(JR) is a maximal abelian subalgebra of OR(JR) and C(K) is a maximal abelian subalgebra of Oγ(K).

AB - 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 OA is simple and purely infinite. Similarly, if the rational function R is restricted to the Julia set JR and the self-similar map γ satisfies the open set condition respectively, then the associated C*-algebras OR(JR) and Oγ(K) are simple and purely infinite. Let ΣA be the associated infinite path space for the 0-1 matrix A, then C(ΣA) is known to be a maximal abelian subalgebra of OA. In this paper we shall show that C(JR) is a maximal abelian subalgebra of OR(JR) and C(K) is a maximal abelian subalgebra of Oγ(K).

KW - Complex dynamical systems

KW - Maximal abelian subalgebras

KW - Self-similar maps

UR - http://www.scopus.com/inward/record.url?scp=85021830354&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85021830354&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2017.06.044

DO - 10.1016/j.jmaa.2017.06.044

M3 - Article

AN - SCOPUS:85021830354

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

ER -