Lyapunov exponents and regularity of invariant foliations for partially hyperbolic diffeomophisms on $\mathbb T^3$

Abstract

We briefly survey some of the recent results concerning the metric behavior of the invariant foliations for a partially hyperbolic on a three-dimensional manifold and propose a conjecture to characterize atomic behavior for conservative partially hyperbolic homotopic to Anosov (DA) on $\mathbb T^3$. On the other hand we prove that if one of the invariant foliations (stable, center or unstable) of a conservative DA on $\mathbb T^3$ is $C^1$ and transversely absolutely continuous with bounded Jacobians the Lyapunov exponent on this direction is defined everywhere and constant. If the center foliation is this foliaion then the DA diffeomophism is smoothly conjugated to a linear Anosov, in particular Anosov. Another consequence of the main theorem is that it does not exist a conservative Ma\~n\'e's example.