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Diffeomorphisms with positive metric entropy
A. Avila1, 2; S. Crovisier3 ; A. Wilkinson4 
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, Volume 124 (2016), pp. 319-347
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We obtain a dichotomy for C1-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.

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DOI: 10.1007/s10240-016-0086-4

A. Avila 1, 2; S. Crovisier 3; A. Wilkinson 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 
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A. Avila; S. Crovisier; A. Wilkinson. Diffeomorphisms with positive metric entropy. Publications Mathématiques de l'IHÉS, Volume 124 (2016), pp. 319-347. doi: 10.1007/s10240-016-0086-4

[AC] F. Abdenur; S. Crovisier Transitivity and topological mixing for C1 diffeomorphisms, Essays in Mathematics and Its Applications (2012), pp. 1-16 | MR | Zbl | DOI

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

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

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

[ABW] A. Avila; J. Bochi; A. Wilkinson Nonuniform center bunching and the genericity of ergodicity among C1 partially hyperbolic symplectomorphisms, Ann. Sci. Éc. Norm. Supér., Volume 42 (2009), pp. 931-979 | MR | Zbl | Numdam | DOI

[ACW] A. Avila, S. Crovisier and A. Wilkinson, C1-density of stable ergodicity, in preparation, | arXiv | MR

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

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

[Boc2] J. Bochi C1-Generic symplectic diffeomorphisms: partial hyperbolicity and zero centre Lyapunov exponents, J. Inst. Math. Jussieu, Volume 9 (2010), pp. 49-93 | MR | DOI | Zbl

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

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

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

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

[BDP] C. Bonatti; L. Diaz; E. Pujals A C1-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources, Ann. Math., Volume 158 (2003), pp. 355-418 | MR | DOI | Zbl

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

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

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

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

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

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

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

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

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

[T1] A. Tahzibi C1-Generic Pesin’s entropy formula, C. R. Acad. Sci. Paris, Volume 335 (2002), pp. 1057-1062 | MR | DOI | Zbl

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