Non-unique Ergodicity, Observers’ Topology and the Dual Algebraic Lamination for R-trees

Let T be an R-tree with a very small action of a free group FN which has dense orbits. Such a tree T or its metric completion T are not locally compact. However, if one adds the Gromov boundary ∂T to T , then there is a coarser observers’ topology on the union T ∪ ∂T , and it is shown here that this union, provided with the observers’ topology, is a compact space b T obs. To any R-tree T as above a dual lamination L2(T ) has been associated in [CHLII]. Here we prove that, if two such trees T0 and T1 have the same dual lamination L(T0) = L(T1), then with respect to the observers’ topology the two trees have homeomorphic compactifications: b T obs 0 = b T obs 1 . Furthermore, if both T0 and T1, say with metrics d0 and d1, respectively, are minimal, this homeomorphism restricts to an FN equivariant bijection T0 → T1, so that on the identified set T0 = T1 one obtains a well defined family of metrics λd1 + (1− λ)d0. We show that for all λ ∈ [0, 1] the resulting metric space Tλ is an R-tree.