Pugh and Shub have conjectured that essential accessibility implies ergodicity for a C 2 , partially hyperbolic, volume-preserving diffeomorphism. We prove this conjecture under a mild center bunching assumption, which is satisfied in particular by all partially hyperbolic systems with 1-dimensional center bundle. We also obtain ergodicity results for C 1Cı partially hyperbolic systems.

We study geodesic flows over compact rank 1 manifolds and prove that sufficiently regular potential functions have unique equilibrium states if the singular set does not carry full pressure. In dimension 2, this proves uniqueness for scalar multiples of the geometric potential on the interval (−∞, 1), which is optimal. In higher dimensions, we obtain the same result on a neighborhood of 0, and give examples where uniqueness holds on all of R. For general potential functions ϕ, we prove that the pressure gap holds whenever ϕ is locally constant on a neighborhood of the singular set, which allows us to give examples for which uniqueness holds on a C 0 -open and dense set of Hölder potentials.

We prove that a diffeomorphism possessing a homoclinic point with a topological crossing (possibly with infinite order contact) has positive topological entropy, along with an analogous statement for heteroclinic points. We apply these results to study area-preserving perturbations of area-preserving surface diffeomorphisms possessing homoclinic and double heteroclinic connections. In the heteroclinic case, the perturbed map can fail to have positive topological entropy only if the perturbation preserves the double heteroclinic connection or if it creates a homoclinic connection. In the homoclinic case, the perturbed map can fail to have positive topological entropy only if the perturbation preserves the connection. These results significantly simplify the application of the Poincarέ-Arnold-Melnikov-Sotomayor method. The results apply even when the contraction and expansion at the fixed point is subexponential.

Let M be a complete Riemannian manifold of bounded nonpositive sectional curvature and finite volume. We construct a topological Tits building A(~I) associated to the universal cover of M. If IV[ is irreducible and rank (M) >I 2, we show that A(~I) is a building canonically associated with a Lie group and hence that M is locally symmetric. I N T R O D U C T I O NLet M be a complete connected Riemannian manifold of bounded nonpositive sectional curvature and finite volume. For any geodesic y, let rank-( be the dimension of the space of parallel Jacobi fields along y. Let rank IV[ be the minimum of the ranks of all geodesics. This definition and the basic structure of such manifolds M with rank M >/ 2 were discussed in [BBE]