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

ABSTRACT

We consider a hyperbolic automorphism $A:{\mathbb{T}}^3\to {\mathbb{T}}^3$ 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 ${\mathbb{T}}^3$. 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.

KEYWORDS:

• partially hyperbolic diffeomorphism
• Anosov diffeomorphism
• SRB measure
• Gibbs u-measure

AMS Subject Classification:

• Primary 37D20
• Secondary 37M05

Acknowledgments

A.G. would like to thank Yakov Pesin who introduced him to questions in the spirit of our Conjecture 1.2 in his 2004 dynamics course. Also, A.G. would like to thank Aleksey Gogolev who performed initial numerical experiments and created the first set of beautiful pictures back in 2008. During final stages of preparation of this paper, discussions with Federico Rodriguez Hertz were very useful. We would like to acknowledge helpful feedback from Dmitry Dolgopyat, Yi Shi, and Rafael Potrie. We also acknowledge helpful comments provided by the referee.

Notes

1 Pesin and Sinai used a stronger definition that is equivalent to the one we give here, see [Bonatti et al. 05, Chapter 11].

2 The first author believes that non-trivial trapping regions exist for some point-wise partially hyperbolic $f:{\mathbb{T}}^3\to {\mathbb{T}}^3$ in the homotopy class of A. If so, it would be interesting to investigate the structure of W^uu_f and how the bifurcation happens.

3 It was suggested to us by Dmitry Dolgopyat that this question also makes sense in higher dimensions if one additionally assumes that f is accessible (or considers u-measures supported on an accessibility class).

4 However, for transitive Anosov diffeomorphisms mixing implies that fⁿ_*ν^u converges to the SRB measure, where ν^u is Lebesgue measure on an unstable plaque. This was explained to us by F. Rodriguez Hertz.

5 We use ‖Δq‖ = 10^{− 7}. With such step size, tests similar to ones in Section 3.1.1 give an upper bound of 10^{− 6} on the precision for weight values.

Additional information

Funding

A.G. was partially supported by the NSF grant DMS-1204943. I.M. and A.N.K. gratefully acknowledge NSF support (Award no. DMR-1410514).