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 study how physical measures vary with the underlying dynamics in the open class of C r , r > 1, strong partially hyperbolic diffeomorphisms for which the central Lyapunov exponents of every Gibbs u-state is positive. If transitive, such a diffeomorphism has a unique physical measure that persists and varies continuously with the dynamics.A main ingredient in the proof is a new Pliss-like Lemma which, under the right circumstances, yields frequency of hyperbolic times close to one. Another novelty is the introduction of a new characterization of Gibbs cu-states. Both of these may be of independent interest.The non-transitive case is also treated: here the number of physical measures varies upper semi-continuously with the diffeomorphism, and physical measures vary continuously whenever possible.Résumé. Nousétudions comment les mesures physiques varient avec la dynamique sousjacente, dans la classe ouverte des difféomorphismes C r , r > 1, fortement partiellement hyperboliques pour lesquelles les exposants de Lyapunov centraux de tout u-état de Gibbs sont positifs. Lorsque transitifs, de tels difféomorphismes possédent une unique mesure physique qui persiste et varie continment avec la dynamique.Un des ingrédients principaux de la preuve est un nouveau lemme de type Pliss qui, appliqué dans le contexte adéquate, implique que la fréquence des temps hyperboliques est proche de un. Une autre nouveauté est l'introduction d'une nouvelle caractérisation des cu-états de Gibbs. Chacun de ses deux aspects ayant leur propre intérêt.Le cas non transitif est aussi traité : dans ce contexte, le nombre de mesures physiques est une fonction semi-continue supérieure du difféomorphisme, et les mesures physiques varient continument sous des hypothèses naturelles.

Abstract. We study ergodic properties of partially hyperbolic systems whose central direction is mostly contracting. Earlier work of Bonatti and Viana (2000) about existence and finitude of physical measures is extended to the case of local diffeomorphisms. Moreover, we prove that such systems constitute a C 2 -open set in which statistical stability is a dense property. In contrast, all mostly contracting systems are shown to be stable under small random perturbations.

Abstract. It is shown that if a non-invertible area preserving local homeomorphism on T 2 is homotopic to a linear expanding or hyperbolic endomorphism, then it must be topologically transitive. This gives a complete characterization, in any smoothness category, of those homotopy classes of conservative endomorphisms that consist entirely of transitive maps.