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March  2004, 11(2&3): 489-516. doi: 10.3934/dcds.2004.11.489

Polynomial growth of the derivative for diffeomorphisms on tori

Krzysztof Frączek 1,

1. 

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

Received  August 2002 Revised  February 2004 Published  June 2004

We consider area--preserving zero entropy ergodic diffeomorphisms on tori. We classify such diffeomorphisms for which the sequence {$Df^n$} has a polynomial growth on the $3$-torus: they are necessary of the form

$\mathbb T^3\quad (x_1,x_2,x_3)\mapsto (x_1+\alpha,\varepsilon x_2+\beta(x_1),x_3+\gamma(x_1,x_2))\in\mathbb T^3,

where $\varepsilon =\pm 1$. We also indicate why there is no $4$-dimensional analogue of the above result. Random diffeomorphisms on the $2$-torus are studied as well.

Keywords: polynomial growth of the derivative, random diffeomorphisms., Area-preserving diffeomorphisms.
Mathematics Subject Classification: 37A05, 37C05, 37C4.
Citation: Krzysztof Frączek. Polynomial growth of the derivative for diffeomorphisms on tori. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 489-516. doi: 10.3934/dcds.2004.11.489
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