Central Lyapunov exponent of partially hyperbolic diffeomorphisms of   $\mathbb{T}^3$
  In this paper we construct some "pathological" volume preserving partially hyperbolic diffeomorphisms on $\toro{3}$ such that their behaviour in small scales in the central direction (Lyapunov exponent) is opposite to the behavior of their linearization. These examples are isotopic to Anosov. We also get partially hyperbolic diffeomorphisms isotopic to Anosov (consequently with non-compact central leaves) with zero central Lyapunov exponent at almost every point. 
Gabriel Ponce and Ali Tahzibi
math.DS