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

TL;DR

This paper numerically investigates Gibbs u-measures for partially hyperbolic diffeomorphisms on the 3-torus, revealing the density of strong unstable manifolds and convergence to SRB measures.

Contribution

It provides the first numerical evidence of the density of strong unstable manifolds and the convergence of push-forward measures to SRB measures in this setting.

Findings

Strong unstable manifold appears dense in -torus

Push-forwards of Lebesgue measure converge to SRB measure

Numerical evidence supports theoretical conjectures

Abstract

We consider a hyperbolic automorphism $A\colon\mathbb T^3\to\mathbb T^3$ 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.