On a surface of negative curvature, we study the distribution of closed geodesics which are not far from minimizing length in their homology classes. These geodesics do not have the statistical behaviour obtained by Margulis for ‘generic’ geodesics and later extended by Lalley to geodesics satisfying certain homological constraints. We cannot use the usual techniques of ‘thermodynamic formalism for Anosov flows’, but we study the problem using ideas from Aubry–Mather theory, as well as some facts of low-dimensional topology.