Smooth Stable Foliations of Anosov Diffeomorphisms

Ruihao Gu

doi:10.5802/crmath.679

Article de recherche - Systèmes dynamiques

Smooth Stable Foliations of Anosov Diffeomorphisms
[Foliations stables et lisses des difféomorphismes d’Anosov]

Ruihao Gu ¹

¹ Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, Peoples Republic of China

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1763-1771.

Abstract (VO)
Résumé

In this paper, we focus on the rigidity of $C^{2 +}$ -smooth codimension-one stable foliations of Anosov diffeomorphisms. Specifically, we show that if the regularity of these foliations is slightly bigger than 2, then they will have the same smoothness of the diffeomorphisms.

Dans cet article, nous nous concentrons sur la rigidité des foliations stables de codimension un des difféomorphismes d’Anosov en $C^{2 +}$ . Plus précisément, nous montrons que si la régularité de ces foliations est légèrement supérieure à 2, alors elles auront la même régularité que les difféomorphismes.

Reçu le : 2023-10-30
Révisé le : 2024-07-10
Accepté le : 2024-08-18
Publié le : 2024-11-25

Zbl

DOI : 10.5802/crmath.679

Classification : 37C05, 37C15, 37D20
Keywords: Anosov diffeomorphism, stable foliation, rigidity
Mots-clés : difféomorphisme d’Anosov, foliation stables, rigidité

Affiliations des auteurs :

Ruihao Gu ¹

¹ Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, Peoples Republic of China

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Smooth {Stable} {Foliations} of {Anosov} {Diffeomorphisms}},
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Ruihao Gu. Smooth Stable Foliations of Anosov Diffeomorphisms. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1763-1771. doi : 10.5802/crmath.679. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.679/

Bibliographie
Cité par

[1] J. An; S. Gan; R. Gu; Y. Shi Rigidity of Stable Lyapunov Exponents and Integrability for Anosov Maps, Commun. Math. Phys., Volume 402 (2023) no. 3, pp. 2831-2877 | DOI | MR | Zbl

[2] D. V. Anosov Geodesic flows on closed Riemannian manifolds of negative curvature, Tr. Mat. Inst. Steklova, Volume 90 (1967), pp. 1-209 | MR | Zbl

[3] R. de la Llave Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems, Commun. Math. Phys., Volume 150 (1992) no. 2, pp. 289-320 | DOI | MR | Zbl

[4] L. Flaminio; A. Katok Rigidity of symplectic Anosov diffeomorphisms on low-dimensional tori, Ergodic Theory Dyn. Syst., Volume 11 (1991) no. 3, pp. 427-441 | DOI | MR | Zbl

[5] J. Franks Anosov diffeomorphisms, Global Analysis (Berkeley, Calif., 1968) (Proceedings of Symposia in Pure Mathematics), Volume 14, American Mathematical Society (1970), pp. 61-93 | MR | Zbl

[6] A. Gogolev; M. Guysinsky $C^{1}$ -differentiable conjugacy of Anosov diffeomorphisms on three dimensional torus, Discrete Contin. Dyn. Syst., Volume 22 (2008) no. 1-2, pp. 183-200 | DOI | MR | Zbl

[7] É. Ghys Rigidité différentiable des groupes fuchsiens, Publ. Math., Inst. Hautes Étud. Sci., Volume 78 (1993), pp. 163-185 | DOI | Numdam | MR | Zbl

[8] S. Hurder; A. Katok Differentiability, rigidity and Godbillon-Vey classes for Anosov flows, Publ. Math., Inst. Hautes Étud. Sci., Volume 72 (1990), pp. 5-61 | DOI | Numdam | MR | Zbl

[9] M. W. Hirsch; C. Pugh Smoothness of horocycle foliations, J. Differ. Geom., Volume 10 (1975), pp. 225-238 | DOI | MR | Zbl

[10] J.-L. Journé A regularity lemma for functions of several variables, Rev. Mat. Iberoam., Volume 4 (1988) no. 2, pp. 187-193 | DOI | MR | Zbl

[11] Y. Katznelson; D. Ornstein The differentiability of the conjugation of certain diffeomorphisms of the circle, Ergodic Theory Dyn. Syst., Volume 9 (1989) no. 4, pp. 643-680 | DOI | MR | Zbl

[12] K. Khanin; A. Teplinsky Herman’s theory revisited, Invent. Math., Volume 178 (2009) no. 2, pp. 333-344 | DOI | MR | Zbl

[13] A. Manning There are no new Anosov diffeomorphisms on tori, Am. J. Math., Volume 96 (1974), pp. 422-429 | DOI | MR | Zbl

[14] S. E. Newhouse On codimension one Anosov diffeomorphisms, Am. J. Math., Volume 92 (1970), pp. 761-770 | DOI | MR | Zbl

[15] Y. B. Pesin Lectures on partial hyperbolicity and stable ergodicity, Zürich Lectures in Advanced Mathematics, European Mathematical Society, 2004, vi+122 pages | DOI | MR | Zbl

[16] C. Pugh; M. Shub; A. Wilkinson Hölder foliations, Duke Math. J., Volume 86 (1997) no. 3, pp. 517-546 | DOI | MR | Zbl

[17] F. Rodriguez Hertz Stable ergodicity of certain linear automorphisms of the torus, Ann. Math., Volume 162 (2005) no. 1, pp. 65-107 | DOI | MR | Zbl

[18] W. M. Schmidt Diophantine approximation, Lecture Notes in Mathematics, 785, Springer, 1980, x+299 pages | MR | Zbl

[19] S. Smale Differentiable dynamical systems, Bull. Am. Math. Soc., Volume 73 (1967), pp. 747-817 | DOI | MR | Zbl

[20] R. Saghin; J. Yang Lyapunov exponents and rigidity of Anosov automorphisms and skew products, Adv. Math., Volume 355 (2019), 106764, 45 pages | DOI | MR | Zbl

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