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We study the map which associates to a current its support (which is a lamination). We show that this map is Out( F   N  )‐equivariant, not injective, not surjective and not continuous. However it is semi‐continuous and almost surjective in a suitable sense. Given an ℝ‐tree  T  (with dense orbits) in the boundary of outer space and a current μ carried by the dual lamination of  T , we define a dual pseudo‐distance  d  μ  on  T . When the tree and the current come from a measured geodesic lamination on a surface with boundary, the dual distance is the original distance of the tree  T . In general, such a good correspondence does not occur. We prove that when the tree  T  is the attractive fixed point of a non‐geometric irreducible, with irreducible powers, outer automorphism, the dual lamination of  T  is uniquely ergodic and the dual distance  d  μ  is either zero or infinite throughout  T .

ℝ‐trees and laminations for free groups III: currents and dual ℝ‐tree metrics