Abstract. 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 1980’s. 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), arxiv.org/abs/2211.10874