Conditional Proof of the Ergodic Conjecture for Falling Ball Systems

TL;DR

This paper provides a conditional proof that most falling ball systems under gravity are ergodic and hyperbolic, assuming certain transversality conditions between singularities and invariant manifolds.

Contribution

It offers a new conditional proof of Wojtkowski's Ergodicity Conjecture for 1D falling ball systems under specific transversality assumptions.

Findings

Almost every system is ergodic and hyperbolic under the assumptions.

The proof relies on transversality between singularities and invariant manifolds.

It advances understanding of ergodic properties in falling ball systems.

Abstract

In this paper we present a conditional proof of Wojtkowski's Ergodicity Conjecture for the system of 1D perfectly elastic balls falling down in a half line under constant gravitational acceleration. Namely, we prove that almost every such system is (completely hyperbolic and) ergodic, by assuming the transversality between different singularities and between singularities and stable (unstable) invariant manifolds.