Upgrading Local Ergodic Theorem for planar semi-dispersing billiards

Local Ergodic Theorem (also known as ‘Fundamental Theorem’) gives sufficient conditions under which a phase point has an open neighborhood that belongs (mod 0) to one ergodic component. This theorem is a key ingredient of many proofs of ergodicity for billiards and, more generally, for smooth hyperbolic maps with singularities. However the proof of that theorem relies upon a delicate assumption (Chernov-Sinai Ansatz), which is difficult to check for some physically relevant models, including gases of hard balls. Here we give a proof of Local Ergodic Theorem for two dimensional billiards without using Ansatz.