OnC 2-diffeomorphisms of the circle which are of type III1

For every irrational number [0, 1) which is not of constant type we construct aC²-diffeomorphism of the circle with rotation number which is of type III₁. This diffeomorphism can be chosen arbitrarily close to the rotationR. Our methods also allow us to construct, for every Liouville number [0, 1), aC-diffeomorphism of the circle with rotation number which is of type III₁.