Almost sure convergence of differentially positive systems on a globally orderable manifold

Differentially positive systems are nonlinear systems whose linearization along trajectories preserves a cone field on a smooth Riemannian manifold. The structures of cone field come from general relativity and Lie theory. We prove that on a globally orderable manifold, the set of convergent points has full Riemann-Lebesgue measure, thus establishing almost sure convergence. This result thereby resolves a measure-theoretic form of the conjecture posed by Forni and Sepulchre in 2016 for such manifolds.