# The core of C*-algebras associated with circle maps

By Benjamin Randeris Johannesen
PhD Dissertations
April 2017
Abstract:

Let $$\phi\colon \mathbb{T}\to\mathbb{T}$$ be any (surjective) continuous and piecewise monotone circle map. We consider the principal and locally compact Hausdorff étale groupoid $$R_\phi^+$$ from [50]. Already $$C^\ast_r(R_\phi^+)$$ is a unital separable direct limit of Elliott--Thomsen building blocks. A characterization of simplicity of $$C^\ast_r(R_\phi^+)$$ is given assuming surjectivity in addition. We also prove that $$C^\ast_r(R_\phi^+)$$ has a unique tracial state and real rank zero when simple. As a consequence $$C^\ast_r(R_\phi^+)$$ has slow dimension growth in the sense of [36] when simple. This means that $$C^\ast_r(R_\phi^+)$$ are classified by their graded ordered K-theory due to [58]. We compute $$K_0(C^\ast_r(R_\phi^+))$$ for a subclass of circle maps. In general $$K_1(C^\ast_r(R_\phi^+)) \simeq \mathbb{Z}$$. 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 $$C^\ast_r(R_\phi^+)$$ 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.