Gromov’s Amenable Localization and Geodesic Flows

Let M be a compact smooth Riemannian n-manifold with boundary. We combine Gromov’s amenable localization technique the Poincaré duality to study traversally generic geodesic flows on SM, space of spherical tangent bundle. Such generate stratifications governed by rich universal combinatorics. The stratification reflects ways in which flow trajectories are boundary $$\partial (SM)$$ . Specifically, we get lower estimates numbers connected components these flow-generated strata any given codimension k terms normed homology $$H_k(M; \mathbb R)$$ and $$H_k(DM; , where $$DM = M\cup _{\partial M} M$$ denotes double M. norms here simplicial semi-norms homology. more complex metric is, numerous SM S(DM) are. It turns out that spaces form obstructions existence globally k-convex metrics also prove knowing scattering map makes it possible reconstruct stratified topological type geodesics, as well amenably localized operators SM.