On a compact hyperbolic surface — negative curvature everywhere, no boundary, no holes opening to infinity — the geodesic flow is expansive. Any two orbits that stay close forever must be the same orbit. The negative curvature forces nearby trajectories apart, and compactness prevents them from escaping to a region where the separation stalls. This is a foundational result in hyperbolic dynamics: negative curvature plus compactness equals orbit distinguishability.
Sergi Burniol Clotet and Françoise Dal'Bo (arXiv:2603.24310, March 2026) show that a single cusp breaks this. A cusp is a point where the surface extends to infinite area through a narrowing funnel — like the end of a trumpet that never quite closes. The surface is still hyperbolic everywhere, still negatively curved, still has the same local geometry that forces orbits apart. But the cusp provides a region where orbits can travel far without diverging, because the funnel's geometry compresses the available space.
The proof uses strong-stable sets. In the cusp region, geodesics heading into the funnel can run alongside each other for arbitrarily long times before eventually diverging. Two distinct orbits can shadow each other to any desired precision by spending time deep in the cusp. The cusp functions as a dynamical trap — not trapping orbits permanently, but lending them enough time together that the expansiveness criterion fails.
The result is clean: any hyperbolic surface with at least one cusp is not expansive. The property is not degraded or weakened — it is lost entirely. One cusp is enough. The infinite funnel, despite being measure-zero in a certain sense, destroys a global dynamical property that the entire rest of the surface supports.
The structural lesson: expansiveness is not a local property. Every point on the surface contributes the same local dynamics — the curvature is uniformly negative. But the global topology provides a refuge where those dynamics cannot enforce separation. The cusp is geometrically small but dynamically decisive. A property that holds everywhere locally can fail globally because of a single topological feature.