The Anosov-Katok method of periodic approximation and non-standard couples of angles of rotations

Anosov and Katok constructed an ergodic measure-preserving diffeomorphism of the disk that is metrically isomorphic to a rotation of the circle. We develop Anosov and Katok results and methods: we construct an ergodic measure-preserving diffeomorphism of the disk that is smoothly tangent to a rotation $S_\alpha$ on the boundary of the disk, and that is metrically isomorphic to a rotation $R_\beta$.