The amount of nonhyperbolicity for partially hyperbolic diffeomorphisms

We study the amount of nonhyperbolicity within a broad class of (nonhyperbolic) partially hyperbolic diffeomorphisms with a one-dimensional center. For that, we focus on the center Lyapunov exponent and the entropy of its level sets. We show that these entropies vary continuously and can be expressed in terms of restricted variational principles. In this study, no dynamical coherence is required. Of particular interest is the case where the exponent is zero. To study this level set, we construct a compact set foliated by curves tangent to the central direction. Within this set, the entropy attains the maximal possible (and positive) value. Moreover, finite-time Lyapunov exponents converge uniformly to zero. In this construction, we introduce a mechanism to concatenate center curves. The class studied consists of those robustly transitive diffeomorphisms that have a pair of blender-horseshoes with different types of hyperbolicity and possess minimal strong stable and unstable foliations. This classes includes flow-type and circle-fibered diffeomorphisms as well as some derived from Anosov diffeomorphisms. It also includes the so-called anomalous examples which are dynamically incoherent.