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Aarhus Universitets segl

The core of C*-algebras associated with circle maps

by Benjamin Randeris Johannesen
PhD Dissertations April 2017

Let ϕ:TT be any (surjective) continuous and piecewise monotone circle map. We consider the principal and locally compact Hausdorff étale groupoid R+ϕ from [50]. Already Cr(R+ϕ) is a unital separable direct limit of Elliott--Thomsen building blocks. A characterization of simplicity of Cr(R+ϕ) is given assuming surjectivity in addition. We also prove that Cr(R+ϕ) has a unique tracial state and real rank zero when simple. As a consequence Cr(R+ϕ) has slow dimension growth in the sense of [36] when simple. This means that Cr(R+ϕ) are classified by their graded ordered K-theory due to [58]. We compute K0(Cr(R+ϕ)) for a subclass of circle maps. In general K1(Cr(R+ϕ))Z. A counterexample yields non-semiconjugate circle maps with isomorphic K-theory.

We give a classification of transitive critically finite circle maps up to conjugacy. This class of circle maps contains the surjective circle maps for which Cr(R+ϕ) is simple. A transitive circle map is always conjugate to a uniformly piecewise linear circle map. We offer a constructive approach to this fact, which also implies a uniqueness result.

Format available: PDF (1 MB)
Thesis advisor: Klaus Thomsen