A numerical study of Gibbs $u$-measures for partially hyperbolic diffeomorphisms on ${\mathbb{T}}^3$

Abstract

We consider a hyperbolic automorphism $A:{\mathbb{T}}^3\to {\mathbb{T}}^3$ of the 3-torus whose 2-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 ${\mathbb{T}}^3$. Further, we calculate push-forwards of the Lebesgue measure on a local strong unstable manifold. These numeric data indicate that the sequence of push-forwards converges to the SRB measure.

12 pages, 12 figures