Anosov’s Ergodic theorem for uniformly hyperbolic diffeomorphisms : the stable and unstable foliations (a) cover the whole space, (b) are absolutely continuous, (c) are of uniform size, and (d) are uniformly transverse to each other.

Sinai gave a systematic method to prove the ergodicity for hyperbolic systems with singularities. Under some mild conditions, Katok and Strelcyn proved that those two foliations cover a full measure set of the space and are still absolutely continuous. However, their leaves can be arbitrarily short, and the angle between them can be arbitrarily small. Assume the singularity sets of the iterates are regular. The Sinai Theorem states that local ergodicity holds if the stable and unstable cones are relatively small while the separation of the two cones is relatively not small. Then the short stable leaves and the short unstable leaves can be used to obtain local ergodic theorem. Assume that there is a continuous invariant cone field on . Then a sufficient condition for both small cones and non-small separation of cones is that , by Liverani and Wojtkowski.