Journal of Modern Dynamics (JMD)

Nonstandard smooth realization of translations on the torus

Pages: 329 - 367, Issue 3, September 2013      doi:10.3934/jmd.2013.7.329

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Mostapha Benhenda - 1 allee Vauban, 92320 Chatillon, France (email)

Abstract: Let $M$ be a smooth compact connected manifold of dimension greater than two, on which there exists a free (modulo zero) smooth circle action that preserves a positive smooth volume. In this article, we construct volume-preserving diffeomorphisms on $M$ that are metrically isomorphic to ergodic translations on the torus of dimension greater than two, where one given coordinate of the translation is an arbitrary Liouville number. To obtain this result, we determine sufficient conditions on translation vectors of the torus that allow us to explicitly construct the sequence of successive conjugacies in Anosov--Katok's method, with suitable estimates of their norm.

Keywords:  Smooth realization, periodic approximation, successive conjugacies, Anosov--Katok.
Mathematics Subject Classification:  Primary: 37C40; Secondary: 37E30.

Received: May 2012;      Revised: August 2013;      Available Online: December 2013.