A Conservative Partially Hyperbolic Dichotomy: Hyperbolicity versus Nonhyperbolic Measures

Abstract: In a conservative and partially hyperbolic three-dimensional setting, we study three representative classes of diffeomorphisms: those homotopic to Anosov (or Derived from Anosov diffeomorphisms), diffeomorphisms in neighborhoods of the time-one map of the geodesic flow on a surface of negative curvature, and accessible and dynamically coherent skew products with circle fibers. In any of these classes, we establish the following dichotomy: either the diffeomorphism is Anosov, or it possesses nonhyperbolic ergodic measures. Our approach is perturbation-free and combines recent advances in the study of stably ergodic diffeomorphisms with a variation of the periodic approximation method to obtain ergodic measures. A key result in our construction, independent of conservative hypotheses, is the construction of nonhyperbolic ergodic measures for sets with a minimal strong unstable foliation that satisfy the mostly expanding property. This approach enables us to obtain nonhyperbolic ergodic measures in other contexts, including some subclasses of the so-called anomalous partially hyperbolic diffeomorphisms that are not dynamically coherent.