Hyperbolicity, heterodimensional cycles and Lyapunov exponents for partially hyperbolic dynamics

Abstract

We prove a dichotomy of C2 partially hyperbolic sets with one-dimensional center direction admitting no zero Lyapunov exponents, either hyperbolicity over the supports of ergodic measures or approximation by a heterodimensional cycle. This provides a partial result to the C1 Palis Conjecture that claims a dichotomy, hyperbolicity or homoclinic bifurcations in a dense subset of the space of C1 diffeomorphisms. Moreover, a theorem of Mane applied in the proof is modified to have an additional property concerning the Hausdorff distance between a periodic orbit and the support of a hyperbolic ergodic measure.