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 aC ²-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.