Polynomial growth of the derivative for diffeomorphisms on tori

Abstract

We consider area--preserving diffeomorphisms on tori with zero entropy. We classify ergodic area--preserving diffeomorphisms of the 3--torus for which the sequence $\{Df^n\}_{n\in{\Bbb N}}$ 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 \[\tor^3\ni(x_1,x_2,x_3)\mapsto (x_1+\alpha,\ep x_2+\beta(x_1),x_3+\gamma(x_1,x_2))\in\tor^3,\] 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.