Non-unique ergodicity, observers' topology and the dual algebraic lamination for $\Bbb R$-trees

Abstract

Let $T$ be an $\R$-tree with a very small action of a free group $F_N$ which has dense orbits. Such a tree $T$ or its metric completion $\bar T$ are not locally compact. However, if one adds the Gromov boundary $\partial T$ to $\bar T$, then there is a coarser \emph{observers' topology} on the union $\bar T \cup \partial T$, and it is shown here that this union, provided with the observers' topology, is a compact space $\Tobs$.

To any $\R$-tree $T$ as above a \emph{dual lamination} $L^{2}(T)$ has been associated in \cite{chl1-II}. Here we prove that, if two such trees $T_{0}$ and $T_{1}$ have the same dual lamination $L^{2}(T_{0}) = L^{2}(T_{1})$, then with respect to the observers' topology the two trees have homeomorphic compactifications: $\Tobszero = \Tobsone$. Furthermore, if both $T_{0}$ and $T_{1}$, say with metrics $d_{0}$ and $d_{1}$, respectively, are minimal, this homeomorphism restricts to an $\FN$-equivariant bijection ${T_{0}} \to {T_{1}}$, so that on the identified set $ {T_{0}} = {T_{1}}$ one obtains a well defined family of metrics $\lambda d_{1}+(1-\lambda)d_{0}$. We show that for all $\lambda \in[0,1]$ the resulting metric space $T_{\lambda}$ is an $\R$-tree.