In the previous chapter we proved that any orientation preserving circle homeomorphism f having irrational rotation number \(\rho \) is semi-conjugate to the rotation of angle \(\rho \) , i.e. there exists a continuous surjective circle map h such that \(h\circ f=r_{\rho}\circ h\) . A natural question arises at this point: is such h an homeomorphism? In other words: is f conjugate to the rotation \(r_{\rho}\) ? The answer is in general negative. To ensure the existence of the conjugacy we will need to impose some smoothness conditions on the function: instead of homeomorphisms we will consider circle diffeomorphisms. The motivation is that in the case of diffeomorphisms we can control how iterates distort the geometry.