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 Λ of any C 1 -generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set Λ.In addition, confirming a claim made by R. Mañé 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 .

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We prove that there is a residual subset I of Diff 1 (M) such that any homoclinic class of a diffeomorphism f ∈ I having saddles of indices α and β contains a dense subset of saddles of index τ for every τ ∈ [α, β] ∩ N. We also derive some consequences from this result about the Lyapunov exponents of periodic points and the sort of bifurcations inside homoclinic classes of generic diffeomorphisms.

In [23], Mary Rees has constructed a minimal homeomorphism of the 2-torus with positive topological entropy. This homeomorphism f is obtained by enriching the dynamics of an irrational rotation R. We improve Rees construction, allowing to start with any homeomorphism R instead of an irrational rotation and to control precisely the measurable dynamics of f . This yields in particular the following result: Any compact manifold of dimension d ≥ 2 which carries a minimal uniquely ergodic homeomorphism also carries a minimal uniquely ergodic homeomorphism with positive topological entropy.More generally, given some homeomorphism R of a (compact) manifold and some homeomorphism h C of a Cantor set, we construct a homeomorphism f which "looks like" R from the topological viewpoint and "looks like" R × h C from the measurable viewpoint. This construction can be seen as a partial answer to the following realisability question: which measurable dynamical systems are represented by homeomorphisms on manifolds ?AMS classification. 37E30, 37B05, 37B40.