We define a dual lamination for any isometric very small FN‐action on an ℝ‐tree T. We obtain an Out (FN)‐equivariant map from the boundary of the outer space to the space of laminations. This map generalizes the corresponding basic construction for surfaces. It fails to be continuous. We then focus on the case where the tree T has dense orbits. In this case, we give two other equivalent constructions, but of different nature, of the dual lamination.

This paper is the first of a sequence of three papers, where the concept of a real tree dual to a measured geodesic lamination in a hyperbolic surface is generalized to arbitrary real trees provided with a (very small) action of a free group by isometries. Laminations for free groups are defined with care in three different approaches: algebraic laminations, symbolic laminations, and laminary languages. The topology on the space of laminations and the action of the outer automorphisms group are detailed.

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Let T be a R-tree in the boundary of the Outer Space CV N , with dense orbits. The Q-index of T is defined by means of the dual lamination of T . It is a generalisation of the Poincaré-Lefschetz index of a foliation on a surface. We prove that the Q-index of T is bounded above by 2N − 2, and we study the case of equality. The main tool is to develop the Rips Machine in order to deal with systems of isometries on compact R-trees.Combining our results on the Q-index with results on the classical geometric index of a tree, developed by Gaboriau and Levitt [GL95], we obtain a beginning of classification of trees.

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 .