On diffeomorphisms with polynomial growth of the derivative on surfaces
Tom 99 / 2004
Colloquium Mathematicum 99 (2004), 75-90
MSC: 37A05, 37C05, 37C40.
DOI: 10.4064/cm99-1-8
Streszczenie
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.
