# American Institute of Mathematical Sciences

July  2013, 7(3): 329-367. doi: 10.3934/jmd.2013.7.329

## Nonstandard smooth realization of translations on the torus

 1 1 allee Vauban, 92320 Chatillon, France

Received  May 2012 Revised  August 2013 Published  December 2013

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.
Citation: Mostapha Benhenda. Nonstandard smooth realization of translations on the torus. Journal of Modern Dynamics, 2013, 7 (3) : 329-367. doi: 10.3934/jmd.2013.7.329
##### References:
 [1] D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trans. Moscow Math. Soc., 23 (1970), 1-35.  Google Scholar [2] B. Fayad and M. Saprykina, Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary, Ann. Sci. Éole Norm. Sup. (4), 38 (2005), 339-364. doi: 10.1016/j.ansens.2005.03.004.  Google Scholar [3] B. Fayad, M. Saprykina and A. Windsor, Non-standard smooth realizations of Liouville rotations, Ergodic Theory and Dynam. Systems, 27 (2007), 1803-1818. doi: 10.1017/S0143385707000314.  Google Scholar [4] P. R. Halmos, Lectures on Ergodic Theory, Publications of the Mathematical Society of Japan, No. 3, The Mathematical Society of Japan, 1956.  Google Scholar [5] A. B. Katok and A. M. Stepin, Approximations in ergodic theory, Russ. Math. Surv., 22 (1967), 77-102. doi: 10.1070/RM1967v022n05ABEH001227.  Google Scholar [6] B. Weiss, The isomorphism problem in ergodic theory, Bull. Amer. Math. Soc., 78 (1972), 668-684.  Google Scholar

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##### References:
 [1] D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trans. Moscow Math. Soc., 23 (1970), 1-35.  Google Scholar [2] B. Fayad and M. Saprykina, Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary, Ann. Sci. Éole Norm. Sup. (4), 38 (2005), 339-364. doi: 10.1016/j.ansens.2005.03.004.  Google Scholar [3] B. Fayad, M. Saprykina and A. Windsor, Non-standard smooth realizations of Liouville rotations, Ergodic Theory and Dynam. Systems, 27 (2007), 1803-1818. doi: 10.1017/S0143385707000314.  Google Scholar [4] P. R. Halmos, Lectures on Ergodic Theory, Publications of the Mathematical Society of Japan, No. 3, The Mathematical Society of Japan, 1956.  Google Scholar [5] A. B. Katok and A. M. Stepin, Approximations in ergodic theory, Russ. Math. Surv., 22 (1967), 77-102. doi: 10.1070/RM1967v022n05ABEH001227.  Google Scholar [6] B. Weiss, The isomorphism problem in ergodic theory, Bull. Amer. Math. Soc., 78 (1972), 668-684.  Google Scholar
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