Polynomial growth of the derivative for diffeomorphisms on tori

Krzysztof Frączek

doi:10.3934/dcds.2004.11.489

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2004, Volume 11, Issue 2&3: 489-516. Doi: 10.3934/dcds.2004.11.489

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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

Received: July 31, 2002

Revised: January 31, 2004

Published: August 2004

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Abstract

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.

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Mathematics Subject Classification: 37A05, 37C05, 37C40.

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