Diffeomorphisms with positive metric entropy

Avila, A.; Crovisier, S.; Wilkinson, A.

doi:10.1007/s10240-016-0086-4

Avila, A.^1,² ; Crovisier, S.³ ; Wilkinson, A.⁴

¹ CNRS, IMJ-PRG, UMR 7586, Univ. Paris Diderot, Sorbonne Paris Cité, Sorbonne Universités, UPMC Univ. Paris 06 75013 Paris France
² IMPA, Estrada Dona Castorina 110 Rio de Janeiro Brazil
³ CNRS, Laboratoire de Mathématiques d’Orsay, UMR 8628, Université Paris-Sud 11 91405 Orsay Cedex France
⁴ Department of Mathematics, University of Chicago 5734 S. University Avenue 60637 Chicago IL USA

Publications Mathématiques de l'IHÉS, Tome 124 (2016), pp. 319-347.

Résumé

We obtain a dichotomy for $C^{1}$ -generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. nonuniformly hyperbolic and the splitting into stable and unstable spaces is dominated). This completes a program first put forth by Ricardo Mañé.

MR Zbl

DOI : 10.1007/s10240-016-0086-4

Affiliations des auteurs :

Avila, A. ^{1, 2} ; Crovisier, S. ³ ; Wilkinson, A. ⁴

@article{PMIHES_2016__124__319_0,
     author = {Avila, A. and Crovisier, S. and Wilkinson, A.},
     title = {Diffeomorphisms with positive metric entropy},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {319--347},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {124},
     year = {2016},
     doi = {10.1007/s10240-016-0086-4},
     mrnumber = {3578917},
     zbl = {1362.37017},
     language = {en},
     url = {https://www.numdam.org/articles/10.1007/s10240-016-0086-4/}
}

TY  - JOUR
AU  - Avila, A.
AU  - Crovisier, S.
AU  - Wilkinson, A.
TI  - Diffeomorphisms with positive metric entropy
JO  - Publications Mathématiques de l'IHÉS
PY  - 2016
SP  - 319
EP  - 347
VL  - 124
PB  - Springer Berlin Heidelberg
PP  - Berlin/Heidelberg
UR  - https://www.numdam.org/articles/10.1007/s10240-016-0086-4/
DO  - 10.1007/s10240-016-0086-4
LA  - en
ID  - PMIHES_2016__124__319_0
ER  -

%0 Journal Article
%A Avila, A.
%A Crovisier, S.
%A Wilkinson, A.
%T Diffeomorphisms with positive metric entropy
%J Publications Mathématiques de l'IHÉS
%D 2016
%P 319-347
%V 124
%I Springer Berlin Heidelberg
%C Berlin/Heidelberg
%U https://www.numdam.org/articles/10.1007/s10240-016-0086-4/
%R 10.1007/s10240-016-0086-4
%G en
%F PMIHES_2016__124__319_0

Avila, A.; Crovisier, S.; Wilkinson, A. Diffeomorphisms with positive metric entropy. Publications Mathématiques de l'IHÉS, Tome 124 (2016), pp. 319-347. doi : 10.1007/s10240-016-0086-4. https://www.numdam.org/articles/10.1007/s10240-016-0086-4/

Bibliographie
Cité par

[AC] Abdenur, F.; Crovisier, S. Transitivity and topological mixing for $C^{1}$ diffeomorphisms, Essays in Mathematics and Its Applications (2012), pp. 1-16 | DOI | MR | Zbl

[An1] D. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Tr. Mat. Inst. Steklova, 90 (1967). | MR | Zbl

[Av] Avila, A. On the regularization of conservative maps, Acta Math., Volume 205 (2010), pp. 5-18 | DOI | MR | Zbl

[AB] Avila, A.; Bochi, J. Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms, Trans. Am. Math. Soc., Volume 364 (2012), pp. 2883-2907 | DOI | MR | Zbl

[ABW] Avila, A.; Bochi, J.; Wilkinson, A. Nonuniform center bunching and the genericity of ergodicity among $C^{1}$ partially hyperbolic symplectomorphisms, Ann. Sci. Éc. Norm. Supér., Volume 42 (2009), pp. 931-979 | DOI | Numdam | MR | Zbl

[ACW] A. Avila, S. Crovisier and A. Wilkinson, $C^{1}$ -density of stable ergodicity, in preparation, | arXiv | MR

[BaBo] Baraviera, A.; Bonatti, C. Removing zero Lyapunov exponents, Ergod. Theory Dyn. Syst., Volume 23 (2003), pp. 1655-1670 | DOI | MR | Zbl

[Boc1] Bochi, J. Genericity of zero Lyapunov exponents, Ergod. Theory Dyn. Syst., Volume 22 (2002), pp. 1667-1696 | DOI | MR | Zbl

[Boc2] Bochi, J. $C^{1}$ -Generic symplectic diffeomorphisms: partial hyperbolicity and zero centre Lyapunov exponents, J. Inst. Math. Jussieu, Volume 9 (2010), pp. 49-93 | DOI | MR | Zbl

[BoBo] Bochi, J.; Bonatti, C. Perturbation of the Lyapunov spectra of periodic orbits, Proc. Lond. Math. Soc., Volume 105 (2012), pp. 1-48 | DOI | MR | Zbl

[BV2] Bochi, J.; Viana, M. The Lyapunov exponents of generic volume-preserving and symplectic maps, Ann. Math., Volume 161 (2005), pp. 1423-1485 | DOI | MR | Zbl

[BFP] Bochi, J.; Fayad, B.; Pujals, E. A remark on conservative diffeomorphisms, C. R. Math. Acad. Sci. Paris, Volume 342 (2006), pp. 763-766 | DOI | MR | Zbl

[BC] Bonatti, C.; Crovisier, S. Récurrence et généricité, Invent. Math., Volume 158 (2004), pp. 33-104 | DOI | MR | Zbl

[BDP] Bonatti, C.; Diaz, L.; Pujals, E. A $C^{1}$ -generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources, Ann. Math., Volume 158 (2003), pp. 355-418 | DOI | MR | Zbl

[F] Franks, J. Anosov diffeomorphisms. Global analysis I, Proc. Symp. Pure Math., Volume 14 (1970), pp. 61-93 | DOI | MR | Zbl

[Ka] Katok, A. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., Volume 51 (1980), pp. 137-173 | DOI | Numdam | MR | Zbl

[M] R. Mañé, Oseledec’s theorem from the generic viewpoint, in Proc. Int. Congress of Mathematicians (Warszawa, 1983), vol. 2, pp. 1259–1276. | MR | Zbl

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

[OU] Oxtoby, J.; Ulam, S. Measure-preserving homeomorphisms and metrical transitivity, Ann. Math. (2), Volume 42 (1941), pp. 874-920 | DOI | MR | Zbl

[P] Pesin, Y. Characteristic Lyapunov exponents, and smooth ergodic theory, Usp. Mat. Nauk, Volume 32 (1977), pp. 55-112 (287) | MR | Zbl

[R] Rodriguez-Hertz, M. A. Genericity of nonuniform hyperbolicity in dimension 3, J. Mod. Dyn., Volume 6 (2012), pp. 121-138 | DOI | MR | Zbl

[RRTU] Rodriguez-Hertz, F.; Rodriguez-Hertz, M. A.; Tahzibi, A.; Ures, R. New criteria for ergodicity and nonuniform hyperbolicity, Duke Math. J., Volume 160 (2011), pp. 599-629 | DOI | MR | Zbl

[SW] Shub, M.; Wilkinson, A. Pathological foliations and removable zero exponents, Invent. Math., Volume 139 (2000), pp. 495-508 | DOI | MR | Zbl

[ST] Sun, W.; Tian, X. Dominated splitting and Pesin’s entropy formula, Discrete Contin. Dyn. Syst., Volume 32 (2012), pp. 1421-1434 | DOI | MR | Zbl

[T1] Tahzibi, A. $C^{1}$ -Generic Pesin’s entropy formula, C. R. Acad. Sci. Paris, Volume 335 (2002), pp. 1057-1062 | DOI | MR | Zbl

Cité par Sources :