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 .

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.

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We study the ergodic properties of generic continuous dynamical systems on compact manifolds. As a main result we prove that generic homeomorphisms have convergent Birkhoff averages under continuous observables at Lebesgue almost every point. In spite of this, when the underlying manifold has dimension greater than one, generic homeomorphisms have no physical measures -a somewhat strange result which stands in sharp contrast to current trends in generic differentiable dynamics. Similar results hold for generic continuous maps.To further explore the mysterious behaviour of C 0 generic dynamics, we also study the ergodic properties of continuous maps which are conjugated to expanding circle maps. In this context, generic maps have divergent Birkhoff averages along orbits starting from Lebesgue almost every point.

We prove that given a compact n-dimensional boundaryless manifold M , n 2, there exists a residual subset R of Diff 1 (M) such that if f ∈ R admits a spectral decomposition (i.e., the nonwandering set Ω(f) admits a partition into a finite number of transitive compact sets), then this spectral decomposition is robust in a generic sense (tame behavior). This implies a C 1-generic trichotomy that generalizes some aspects of a two-dimensional theorem of Mañé [Topology 17 (1978) 386-396]. Lastly, Palis [Astérisque 261 (2000) 335-347] has conjectured that densely in Diff k (M) diffeomorphisms either are hyperbolic or exhibit homoclinic bifurcations. We use the aforementioned results to prove this conjecture in a large open region of Diff 1 (M).  2003 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ.-Nous montrons qu'étant donnée une variété compacte M de dimension n, n 2, il existe un sous-ensemble résiduel R de Diff 1 (M) tel que si f ∈ R admet une décomposition spectrale (c'est-à-dire, Ω(f) admet une partition en un nombre fini d'ensembles compacts transitifs), alors cette décomposition spectrale est robuste dans un sens générique. Cela implique une trichotomie générique qui généralise certains aspects d'un théorème bi-dimensionnel de Mañé. Enfin, Palis a conjecturé que dans un sous-ensemble dense de Diff 1 (M), les difféomorphismes ou bien sont hyperboliques, ou bien admettent des bifurcations homoclines. Nous utilisons les résultats précédents pour prouver cette conjecture dans une grande région ouverte de Diff 1 (M).