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Noncommutative Torus from Fibonacci Chains via Foliation

Hyeong-Chai Jeong·Eunsang Kim·Chang-Yeong Lee
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Abstract

We classify the Fibonacci chains (F-chains) by their index sequences and construct an approximately finite dimensional (AF) $C^*$-algebra on the space of F-chains as Connes did on the space of Penrose tiling. The K-theory on this AF-algebra suggests a connection between the noncommutative torus and the space of F-chains. A noncommutative torus, which can be regarded as the $C^*$-algebra of a foliation on the torus, is explicitly embedded into the AF-algebra on the space of F-chains. As a counterpart of that, we obtain a relation between the space of F-chains and the leaf space of Kronecker foliation on the torus using the cut-procedure of constructing F-chains.

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Hyeong-Chai Jeong, Eunsang Kim, Chang-Yeong Lee. 2000-08-22. Noncommutative Torus from Fibonacci Chains via Foliation. https://doi.org/10.1088/0305-4470%2F34%2F31%2F201

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