$\mathbb{R}$-trees, dual laminations, and compact systems of partial isometries

Let $\text{FN}$ be a free group of finite rank $N\ge 2$, and let $T$ be an $\mathbb{R}$-tree with a very small, minimal action of $\text{FN}$ with dense orbits. For any basis $\text{CA}$ of $\text{FN}$ there exists a {\em heart} $K_{\text{CA}}\subset \bar{T}$ (= the metric completion of $T$) which is a compact subtree that has the property that the dynamical system of partial isometries $a_{i} : K_{\CA} \cap a_{i} K_{\CA} \to a_{i}\inv K_{\CA} \cap K_{\CA}$,foreach$a_{i} \in \CA$,definesatree$T_{(K_{\CA}, \CA)}$whichcontainsanisometriccopyof$T$ as minimal subtree.