Ergodic currents dual to a real tree

TL;DR

This paper investigates ergodic currents dual to a specific type of real tree associated with free group actions, establishing bounds on their number, and introduces an induction process to approximate these currents.

Contribution

It provides a bound on the number of ergodic currents dual to a real tree with dense orbits and develops an unfolding induction method for approximation.

Findings

Bound of 3N-5 on the number of ergodic currents

Introduction of unfolding induction for real trees

A criterion for unique ergodicity

Abstract

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.