A NOTE ON RIGIDITY OF ANOSOV DIFFEOMORPHISMS OF THE THREE TORUS

We consider Anosov diffeomorphisms on T-3 such that the tangent bundle splits into three subbundles E-f(s) circle plus E-f(su). We show that if f is C-r, r >= 2, volume preserving, then f is C-1 conjugated with its linear part A if and only if the center foliation F-f(wu) is absolutely continuous and the equality lambda(wu)(f) (x) = lambda(wu)(A), between center Lyapunov exponents of f and A, holds for m a.e. x is an element of T-3. We also conclude rigidity derived from Anosov diffeomorphism, assuming a strong absolute continuity property (Uniform Bounded Density property) of strong stable and strong unstable foliations.