2
$\begingroup$

The question is to classify uniformly hyperbolic invariant sets supporting uniquely ergodic invariant measure. The only examples that I expect are: fixed points, periodic orbits and Cantori(Denjoy minimal set).

If there is no other possibilities, then what can we say about the Cantori? The structure of the Cantori may reflect the hyperbolicity.

Consider geodesic laminations on surfaces of genus greater than 1 with negative curvature. The classification holds. Moreover, for the Cantori case, if we intersect the lamination using a line segment transversally, then list the resulting short intervals in a decreasing order, then lengths of these intervals decay exponentially. So the question is: do these results hold in higher dimensional cases?

Any comments and references are welcome.

$\endgroup$
2
  • 1
    $\begingroup$ I think you should add some additional hypothesis on the set in order to get such a classification, because if not, there are some (not so interesting) examples. For instance, consider a non-trivial hyperbolic set exhibiting a fixed point (e.g a Smale horseshoe) and let $\Lambda$ be the set formed by a fixed point of the horseshoe and a homoclinic orbit associated to this fixed point. Then $\Lambda$ is hyperbolic and uniquely ergodic. $\endgroup$
    – Alejandro
    Jan 14, 2014 at 16:26
  • $\begingroup$ Thank you. In your example, the homoclinic orbit does not support measure. What I am interested is the support of the measure. $\endgroup$
    – John Galt
    Jan 14, 2014 at 22:34

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.