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The space of geodesic currents on a closed surface is a simultaneous generalization of Teichmuller space, the space of measured laminations, and the space of (not necessarily simple) curves. But they form an infinite-dimensional space that can be hard to get a handle on. We characterize geodesic currents in terms of their intersection number with curves: if a functional on curves satisfies a few simple axioms, most notably a smoothing relation on decrease under resolving a crossing, then it is the intersection number with a current, and conversely.

This is joint work with Didac Martinez-Granado. It extends previous work that showed that these curve functionals extend continuously to geodesic currents.