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.