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

Abstract

We consider a hyperbolic automorphism A : T 3 → T 3 of the 3-torus whose 2-dimensional unstable distribution splits into weak and strong unstable subbundles. We unfold A into two oneparameter families of Anosov diffeomorphismsa 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 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 pushforwards converges to the SRB measure.

On the Three-Legged Accessibility Property

“…In the case of skew products we show that our results can be applied to an open and dense set of isometric circle extensions of volume-preserving partially hyperbolic diffeomorphisms. In the case of Anosov diffeomorphisms, we answer positively the following conjecture of Gogolev, Maimon and Kolgomorov [10]. Let A be a hyperbolic automorphism of the 3-torus with three real eigenvalues…”

Robust minimality of strong foliations for DA diffeomorphisms: 𝑐𝑢-volume expansion and new examples

“…The cone families C u , C s and the inequalities (17), (18) between the determinants satisfy the conditions of Lemma 7.8. Then f k is partially hyperbolic provided that k is large enough.…”

“…It is also an open question for the hyperbolic automorphisms of the 3-torus with three different real eigenvalues whether the strong foliation is robustly minimal. See [17] for a discussion on this topic. In this work we are going to study the robust minimality of these foliations for certain three-dimensional torus diffeomorphisms that we introduce below.…”