Unique ergodicity of asynchronous rotations, and application
[Unique ergodicité des rotations asynchrones, et application]
Maucourant, François1
1 Université de Rennes, IRMAR Campus de Beaulieu 35042 Rennes cedex (France)
Annales de l'Institut Fourier, Online first, 27 p.

The main result of this paper is an analogue for a continuous family of tori of Kronecker–Weyl’s unique ergodicity of irrational rotations. We show that the notion corresponding in this setup to irrationality, namely asynchronicity, is satisfied in some homogeneous dynamical systems. This is used to prove the ergodicity of naturals lifts of invariant measures.

Nous étudions sur une famille continue de tores les rotations dites asynchrones, analogues aux rotations irrationnelles sur les tores classiques. Le résultat principal est l’unique ergodicité de ces rotations sur un monoïde adapté. Nous prouvons que la condition d’asynchronicité est vérifiée dans une famille d’exemples issue de la dynamique homogène, ce qui nous permet de déduire l’ergodicité de relevés de certaines transformations dans des fibrés en tores.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3718
Classification : 37A17
Keywords: unique ergodicity, homogeneous dynamics
Mots-clés : unique ergodicité, dynamique homogène

Maucourant, François 1

1 Université de Rennes, IRMAR Campus de Beaulieu 35042 Rennes cedex (France) 
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Maucourant, François. Unique ergodicity of asynchronous rotations, and application. Annales de l'Institut Fourier, Online first, 27 p.

[1] Aaronson, Jon An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, 50, American Mathematical Society, 1997, xii+284 pages | DOI | MR | Zbl

[2] Arnoux, Pierre; Fisher, Albert M. The scenery flow for geometric structures on the torus: the linear setting, Chin. Ann. Math., Ser. B, Volume 22 (2001) no. 4, pp. 427-470 | DOI | MR | Zbl

[3] Borel, Armand Introduction aux groupes arithmétiques, Actualités Scientifiques et Industrielles, 1341, Hermann, 1969, 125 pages | MR | Zbl

[4] Borel, Armand Linear algebraic groups, Graduate Texts in Mathematics, 126, Springer, 1991, xii+288 pages | DOI | MR | Zbl

[5] Coudène, Yves The Hopf argument, J. Mod. Dyn., Volume 1 (2007) no. 1, pp. 147-153 | DOI | MR | Zbl

[6] Coudene, Yves On invariant distributions and mixing, Ergodic Theory Dyn. Syst., Volume 27 (2007) no. 1, pp. 109-112 | DOI | MR | Zbl

[7] Elkies, Noam D.; McMullen, Curtis T. Gaps in nmod1 and ergodic theory, Duke Math. J., Volume 123 (2004) no. 1, pp. 95-139 | DOI | MR | Zbl

[8] Furstenberg, H. Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, 1981, xi+203 pages (M. B. Porter Lectures) | DOI | MR | Zbl

[9] Kleinbock, Dmitry; Shah, Nimish; Starkov, Alexander Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory, Handbook of dynamical systems, Vol. 1A, North-Holland, 2002, pp. 813-930 | DOI | MR | Zbl

[10] Marklof, Jens; Strömbergsson, Andreas The Boltzmann-Grad limit of the periodic Lorentz gas, Ann. Math. (2), Volume 174 (2011) no. 1, pp. 225-298 | DOI | MR | Zbl

[11] Morris, Dave Witte Ratner’s theorems on unipotent flows, Chicago Lectures in Mathematics, University of Chicago Press, 2005, xii+203 pages | MR | Zbl

[12] Shapira, Uri A solution to a problem of Cassels and Diophantine properties of cubic numbers, Ann. Math. (2), Volume 173 (2011) no. 1, pp. 543-557 | DOI | MR | Zbl

[13] Strömbergsson, Andreas An effective Ratner equidistribution result for SL(2,ℝ)⋉ℝ 2 , Duke Math. J., Volume 164 (2015) no. 5, pp. 843-902 | DOI | MR | Zbl

[14] Thomine, Damien Keplerian shear in ergodic theory, Ann. Henri Lebesgue, Volume 3 (2020), pp. 649-676 | DOI | Numdam | MR | Zbl

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