On critical circle homeomorphisms

We prove that an analytic circle homeomorphism without periodic orbits is conjugated to the linear rotation by a quasi-symmetric map if and only if its rotation number is of constant type. Next, we consider automorphisms of quasi-conformal Jordan curves, without periodic orbits and holomorphic in a neighborhood. We prove a "Denjoy theorem" that such maps are conjugated to a rotation on the circle. © 1998, Sociedade Brasileira de Matemática.