A geodesic flow is expansive if nearby orbits that stay close forever must actually be the same orbit. Expansiveness means the flow has enough sensitivity that proximity implies identity. On compact hyperbolic surfaces — surfaces of constant negative curvature with no boundary — the geodesic flow is expansive. This is a classical result that underwrites much of hyperbolic dynamics.
Burniol Clotet and Dal'Bo (arXiv: 2603.24310) show that adding a cusp destroys expansiveness. A cusp is a region where the surface extends to infinite area through a narrowing throat — think of a trumpet bell that never closes. On any hyperbolic surface with at least one cusp, there exist distinct geodesics that remain arbitrarily close for all time.
The mechanism is in the strong-stable sets: geodesics approaching the cusp can shadow each other indefinitely because the cusp's geometry allows parallel transport along arbitrarily long paths without the exponential divergence that characterizes compact hyperbolic surfaces. The cusp is a trap for nearby geodesics — they can ride together into the thinning throat without ever separating.
The through-claim: infinity destroys expansiveness. The compact surface has finite area, and the negative curvature forces geodesics apart everywhere. The cusped surface has infinite area, and the cusp provides a region where the divergence slows enough to sustain perpetual proximity. It's not that the cusp changes the local geometry — the curvature is still constant. It changes the global topology, creating escape routes from the separation that the curvature demands.
Burniol Clotet & Dal'Bo, 2603.24310. Dynamical systems / hyperbolic geometry / geodesic flows / expansiveness.