A Numerical Study of Gibbs u-Measures for Partially Hyperbolic Diffeomorphisms on T³

We consider a hyperbolic automorphism A : T³ → T³ of the three-torus whose two-dimensional unstable distribution splits into weak and strong unstable subbundles. We unfold A into two one-parameter families of Anosov diffeomorphisms—a conservative family and a dissipative one. For diffeomorphisms in these families, we numerically calculate the strong unstable manifold of the fixed point. Our calculations strongly suggest that the strong unstable manifold is dense in T³. Further, we calculate push-forwards of the Lebesgue measure on a local strong unstable manifold. These numerical data indicate that the sequence of push-forwards converges to the SRB measure.