In the 1980s M. P. Wojtkowski introduced an interesting dynamical system of 1D balls

moving in a vertical half-line, colliding with each other and the hard floor elastically,

and falling down under constant gravitation. To avoid the existence of linearly stable

periodic orbits, one assumes that the masses of the particles are decreasing as we

go up in the half line. He conjectured that all these systems are completely hyperbolic

and ergodic.

Complete hyperbolicity of all such systems was shown by N. S. in 1996. Here we

describe a brand new algebraic approach to such systems that enable us to verify all

the conditions of the Local Ergodic Theorem for Dynamical Systems With Invariant

Cone Fields (by Liverani and Wojtkowski) for almost all such falling ball systems, thus

proving their ergodicity, a famous, so far unsolved conjecture of Maciej P. Wojtkowski

from the mid 1980âs.

In the talk special emphasis will be given to some interesting new aspects of the

exploited algebraic approach that made it possible to prove the annoying

transversality condition (the equivalent of the Chernov-Sinai Ansatz for billiards)

assumed in my conditional result [S2022].

[S2022] Simanyi, N.: “Conditional Proof of the Ergodic Conjecture for Falling Ball

Systems”. To appear in Contemporary Mathematics (2022),

https://arxiv.org/abs/2211.10874