# Certain systems of three falling balls satisfy the Chernov-Sinai ansatz

## TL;DR

This paper advances understanding of the ergodic behavior of a specific three-ball falling system by proving the Chernov-Sinai ansatz, identifying abundant expanding points, and linking proper alignment to transversality conditions.

## Contribution

It proves the Chernov-Sinai ansatz for a specific mass ratio in the three falling balls system and clarifies the relation between proper alignment and transversality conditions.

## Key findings

- Proved the Chernov-Sinai ansatz for the system.
- Established the abundance of least expanding points.
- Linked proper alignment condition to Chernov's transversality condition.

## Abstract

The system of falling balls is an autonomous Hamiltonian system with a smooth invariant measure and non-zero Lyapunov exponents almost everywhere. Since almost three decades, the question of ergodicity is still open. The subject of this work is to contribute to the solution of the ergodicity conjecture for three falling balls with a specific mass ratio. The latter is executed in the following three points: First, we prove the Chernov-Sinai ansatz. Second, we prove that there is an abundance of least expanding points and, third, we explain that the proper alignment condition can still be verified and is actually pointwise equivalent to Chernov's transversality condition. It is of special interest, that for the aforementioned specific mass ratio, the configuration space can be unfolded to a billiard table, where the proper alignment condition holds.