In the shadow of mixing properties, the central limit theorem and similar chaotic behavior a dynamical system may experience, dwells the system of falling balls. The description of the latter system comes at hand: Three balls moving along a vertical line, colliding elastically with each other and the lowest ball collides with a rigid floor placed at height zero. It is not known whether this system possesses the aforementioned chaotic properties because it is not even known yet to be ergodic.

The system of falling balls belongs to the family of non-uniformly hyperbolic systems with singularities. In order to establish ergodicity, it is necessary to check five conditions of the celebrated Local Ergodic Theorem. In this talk I will present a proof for one of them, namely, the Chernov-Sinai Ansatz. It is also planned to give the audience an overview of what is known about the remaining conditions.