Geodesic
Diffeomorphisms with positive metric entropy
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

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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.

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 
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     title = {Diffeomorphisms with positive metric entropy},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {319--347},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
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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

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