Nonstandard smooth realization of translations on the torus

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

Nonstandard smooth realization of translations on the torus

Mostapha Benhenda ^1,

1.	1 allee Vauban, 92320 Chatillon, France

Received May 2012 Revised August 2013 Published December 2013

Abstract
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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: successive conjugacies, periodic approximation, Anosov--Katok., Smooth realization.

Mathematics Subject Classification: Primary: 37C40; Secondary: 37E3.

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.
[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.
[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.
[4]	P. R. Halmos, Lectures on Ergodic Theory, Publications of the Mathematical Society of Japan, No. 3, The Mathematical Society of Japan, 1956.
[5]	A. B. Katok and A. M. Stepin, Approximations in ergodic theory, Russ. Math. Surv., 22 (1967), 77-102. doi: 10.1070/RM1967v022n05ABEH001227.
[6]	B. Weiss, The isomorphism problem in ergodic theory, Bull. Amer. Math. Soc., 78 (1972), 668-684.

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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.
[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.
[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.
[4]	P. R. Halmos, Lectures on Ergodic Theory, Publications of the Mathematical Society of Japan, No. 3, The Mathematical Society of Japan, 1956.
[5]	A. B. Katok and A. M. Stepin, Approximations in ergodic theory, Russ. Math. Surv., 22 (1967), 77-102. doi: 10.1070/RM1967v022n05ABEH001227.
[6]	B. Weiss, The isomorphism problem in ergodic theory, Bull. Amer. Math. Soc., 78 (1972), 668-684.

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