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

Abstract

We study the map which associates to a current its support (which is a lamination). We show that this map is Out(FN)-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 dμ 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 dμ is either zero or infinite throughout T.