Conformally natural extensions in view of dynamics

We give an easy description of the barycentric extension of a map of the unit circle to the closed unit disk using some ideas from dynamical systems. We then prove that every circle endomorphism of the unit circle of degree d ≥ 2 (with a topological expansion condition) has a conformally natural extension to the closed unit disk which is real analytic on the open unit disk. If the endomorphism is uniformly quasisymmetric, then the extension is quasiconformal.