Certain systems of three falling balls satisfy the Chernov-Sinai Ansatz

TL;DR

This paper proves the Chernov-Sinai ansatz and the abundance of expanding points for a specific three falling balls system, advancing the understanding of its ergodicity in a Hamiltonian setting.

Contribution

It establishes the Chernov-Sinai ansatz and demonstrates the presence of expanding points for a particular mass ratio, contributing to the ergodicity conjecture.

Findings

Proved the Chernov-Sinai ansatz for the system.

Showed the existence of many expanding points.

Connected the system to a billiard table with proper alignment.

Abstract

The system of falling balls is an autonomous Hamiltonian system with a smooth invariant measure and non-zero Lyapunov exponents almost everywhere. For almost three decades new, the question of its ergodicity remains open. We contribute to the solution of the erogodicity conjecture for three falling balls with a specific mass ratio in the following two points: First, we prove the Chernov-Sinai ansatz. Second, we prove that there is an abundance of sufficiently expanding points. 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.