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 an 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.Dedicated to the memory of R. MañéPartially supported by NSF grant DMS-9704368 and the Sloan Foundation.