On diffeomorphisms with polynomial growth of the derivative on surfaces

Krzysztof Frączek ¹

¹ Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18 87-100 Toruń, Poland

Colloquium Mathematicum, Tome 99 (2004) no. 1, pp. 75-90

Résumé

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.

Détail

DOI : 10.4064/cm99-1-8

Keywords: consider zero entropy infty diffeomorphisms compact connected infty manifolds introduce notion polynomial growth derivative diffeomorphisms study diffeomorphisms which additionally preserve smooth measure manifold admits ergodic diffeomorphism polynomial growth derivative there exists smooth flow fixed point moreover mathop dim nolimits necessarily mathbb diffeomorphism infty conjugate skew product torus