Non-ergodicity of the motion in three dimensional steep repelling   dispersing potentials
  It is demonstrated numerically that smooth three degrees of freedom Hamiltonian systems which are arbitrarily close to three dimensional strictly dispersing billiards (Sinai billiards) have islands of effective stability, and hence are non-ergodic. The mechanism for creating the islands are corners of the billiard domain. 
Anna Rapoport and Vered Rom-Kedar