The counting problem for closed geodesics over negatively curved manifolds, and more generally, for closed orbits of Anosov flows is studied extensively in the literature. However, when the metric is not negatively curved, or when the flow of a vector field is not Anosov, closed geodesics/orbits are not isolated and there are some obstacles for defining a well-behaved count function. We discuss some of these obstructions. In particular, we introduce a function which assigns an integer weight to every compact and open subset of the space of closed geodesics for arbitrary Riemannian metrics over closed manifolds.

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