Nonuniform hyperbolicity for C^1-generic diffeomorphisms
  We study the ergodic theory of non-conservative C^1-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C^1-generic diffeomorphisms are nonuniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set L of any C^1-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set L. In addition, confirming a claim made by R. Man\'e in 1982, we show that hyperbolic measures whose Oseledets splittings are dominated satisfy Pesin's Stable Manifold Theorem, even if the diffeomorphism is only C^1. 
Flavio Abdenur, Christian Bonatti (IMB), Sylvain Crovisier (LAGA)