Linear growth of the derivative for measure-preserving diffeomorphisms

We consider measure-preserving diffeomorphisms of the torus with zero entropy. We prove that every ergodic ${\displaystyle C^1}$-diffeomorphism with linear growth of the derivative is algebraically conjugate to a skew product of an irrational rotation on the circle and a circle ${\displaystyle C^1}$-cocycle. We also show that for no positive β ≠ 1 does there exist an ergodic ${\displaystyle C^2}$-diffeomorphism whose derivative has polynomial growth with degree β.