Polynomial growth of the derivative for diffeomorphisms on tori

We consider area–preserving diffeomorphisms on tori with zero entropy. We classify ergodic area–preserving diffeomorphisms of the 3–torus for which the sequence $\{D{f}^n{\}}_{n\in \mathbb{N}}$ has polynomial growth. Roughly speaking, the main theorem says that every ergodic area–preserving $C^2$–diffeomorphism with polynomial uniform growth of the derivative is $C^2$–conjugate to a 2–steps skew product of the form $${\text{tor}}^3\ni (x_1,x_2,x_3)\mapsto (x_1+\alpha ,\text{ep}{x}_2+\beta (x_1),x_3+\gamma (x_1,x_2))\in {\text{tor}}^3,$$ where $\ep=\pm 1$. We also indicate why there is no 4–dimensional analogue of the above result. Random diffeomorphisms on the 2–torus are studied as well.