Wojtkowski’s ergodic hypothesis: a conjecture for decades?

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),