Noncommutative Torus from Fibonacci Chains via Foliation

TL;DR

This paper explores the relationship between Fibonacci chains and noncommutative tori by constructing an AF-algebra on F-chains and embedding the noncommutative torus within it, revealing deep connections via foliation and K-theory.

Contribution

It introduces a novel AF-algebra on Fibonacci chains and explicitly embeds the noncommutative torus, linking F-chains to foliation structures on the torus.

Findings

AF-algebra on F-chains constructed

Explicit embedding of noncommutative torus achieved

Connection established between F-chains and Kronecker foliation

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.