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Motivated by Bonahon's result for hyperbolic surfaces, we construct ananalogue of the Patterson-Sullivan-Bowen-Margulis map from the Culler-Vogtmannouter space $CV(F_k)$ into the space of projectivized geodesic currents on afree group. We prove that this map is a topological embedding. We also provethat for every $k\ge 2$ the minimum of the volume entropy of the universalcovers of finite connected volume-one metric graphs with fundamental group ofrank $k$ and without degree-one vertices is equal to $(3k-3)\log 2$ and thatthis minimum is realized by trivalent graphs with all edges of equal lengths,and only by such graphs.Comment: An updated versio

The Patterson–Sullivan Embedding and Minimal Volume Entropy for Outer Space