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 db 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. 1. Conformal barycenter Let C denote the complex plane, T 1⁄4 fz A C : jzj 1⁄4 1g the unit circle, D 1⁄4 fz A C : jzj < 1g the open unit disk, and D 1⁄4 DUT the closed unit disk. Let m be a probability measure on T. The barycenter of m, by definition, is