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.