Ergodic currents dual to a real tree

Abstract

Let $T$ be an $\mathbb{R}$-tree with dense orbits in the boundary of outer space. When the free group $\mathbb{F}_{N}$ acts freely on $T$, we prove that the number of projective classes of ergodic currents dual to $T$ is bounded above by $3N-5$. We combine Rips induction and splitting induction to define unfolding induction for such an $\mathbb{R}$-tree $T$. Given a current ${\it\mu}$ dual to $T$, the unfolding induction produces a sequence of approximations converging towards ${\it\mu}$. We also give a unique ergodicity criterion.