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