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In this paper we present a conditional proof of Wojtkowski's ErgodicityConjecture for the system of 1D perfectly elastic balls falling down in a halfline under constant gravitational acceleration. Namely, we prove that almostevery such system is (completely hyperbolic and) ergodic, by assuming thetransversality between different singularities and between singularities andstable (unstable) invariant manifolds.

Conditional Proof of the Ergodic Conjecture for Falling Ball Systems