We prove that every C 1 diffeomorphism away from homoclinic tangencies is entropy expansive, with locally uniform expansivity constant. Consequently, such diffeomorphisms satisfy Shub's entropy conjecture: the entropy is bounded from below by the spectral radius in homology. Moreover, they admit principal symbolic extensions, and the topological entropy and metrical entropy vary continuously with the map. In contrast, generic diffeomorphisms with persistent tangencies are not entropy expansive and have no symbolic extensions.

Abstract. We show that every diffeomorphism with mostly contracting center direction exhibits a geometric-combinatorial structure, which we call skeleton, that determines the number, basins and supports of the physical measures. Furthermore, the skeleton allows us to describe how the physical measure bifurcate as the diffeomorphism changes. In particular, we use this to construct examples with any given number of physical measures, with basins densely intermingled, and to analyse how these measures collapse into each otherthrough explosions of their basins -as the dynamics varies. This theory also allows us to prove that, in the absence of collapses, the basins are continuous functions of the diffeomorphism.

For a class of partially hyperbolic C k , k > 1 diffeomorphisms with circle center leaves we prove the existence and finiteness of physical (or Sinai-Ruelle-Bowen) measures, whose basins cover a full Lebesgue measure subset of the ambient manifold. Our conditions hold for an open and dense subset of all C k partially hyperbolic skew-products on compact circle bundles.Our arguments blend ideas from the theory of Gibbs states for diffeomorphisms with mostly contracting center direction together with recent progress in the theory of cocycles over hyperbolic systems that call into play geometric properties of invariant foliations such as absolute continuity. Recent results show that absolute continuity of the center foliation is often a rigid property among volume preserving systems. We prove that this is not at all the case in the dissipative setting, where absolute continuity can even be robust.

A deep analysis of the Lyapunov exponents, for stationary sequence of matrices going back to Furstenberg [11], for more general linear cocycles by Ledrappier [18] and generalized to the context of non-linear cocycles by Avila and Viana [1], gives an invariance principle for invariant measures with vanishing central exponents. In this paper, we give a new criterium formulated in terms of entropy for the invariance principle and in particular, obtain a simpler proof for some of the known invariance principle results.As a byproduct, we study ergodic measures of partially hyperbolic diffeomorphisms whose center foliation is 1-dimensional and forms a circle bundle. We show that for any such C 2 diffeomorphism which is accessible, weak hyperbolicity of ergodic measures of high entropy implies that the system itself is of rotation type.