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.