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Abstract

We consider zero entropy $C^{\infty }$-diffeomorphisms on compact connected $C^\infty $-manifolds. We introduce the notion of polynomial growth of the derivative for such diffeomorphisms, and study it for diffeomorphisms which additionally preserve a smooth measure. We show that if a manifold $M$ admits an ergodic diffeomorphism with polynomial growth of the derivative then there exists a smooth flow with no fixed point on $M$. Moreover, if $\mathop {\rm dim}\nolimits M=2$, then necessarily $M={\mathbb T}^2$ and the diffeomorphism is $C^{\infty }$-conjugate to a skew product on the $2$-torus.