Ergodic currents dual to a real tree

Thierry Coulbois (1) , Arnaud Hilion (1)

1 I2M - Institut de Mathématiques de Marseille

Let $T$ be an $\R$-tree in the boundary of Outer space with dense orbits. When the free group $\FN$ 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 $\R$-tree $T$. Given a current $\mu$ dual to $T$, the unfolding induction produces a sequence of approximations converging towards $\mu$. We also give a unique ergodicity criterion.

Domaines: Théorie des groupes [math.GR]

Thierry Coulbois, Arnaud Hilion. Ergodic currents dual to a real tree. Ergodic Theory and Dynamical Systems, 2016, 36 (3), pp.745-766. ⟨10.1017/etds.2014.78⟩. ⟨hal-01481866⟩