Alternative determinism principle for topological analysis of chaos

The topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits has proven to be a powerful method, however knot theory can only be applied to three-dimensional systems. Still, the core principles upon which this approach is built--determinism and continuity--apply in any dimension. We propose an alternative framework in which these principles are enforced on triangulated surfaces rather than curves, and we show that in dimension 3 our approach numerically predicts the correct topological entropies for periodic orbits of the horseshoe map.