A conditional proof of the non-contraction property for N falling balls

TL;DR

This paper proves the non-contraction property for Wojtkowski's falling balls system under certain conditions, advancing understanding of its ergodic behavior and providing a new approach to its analysis.

Contribution

It establishes a conditional proof of non-contraction for the system, and demonstrates ergodicity for a specific case with three degrees of freedom.

Findings

Non-contraction property proven under strict unboundedness assumption

Configuration space can be unfolded to a billiard table satisfying proper alignment

Ergodicity shown for a particular mass ratio case

Abstract

Wojtkowski's system of $N$, $N \geq 2$, falling balls is a nonuniformly hyperbolic smooth dynamical system with singularities. It is still an open question whether this system is ergodic. We contribute towards an affirmative answer, by proving the non-contraction property, conditioned by the assumption of strict unboundedness. For a certain mass ratio the configuration space can be unfolded to a billiard table where the daunting proper alignment condition is satisfied. We prove, that the aforementioned unfolded system with three degrees of freedom is ergodic.