Ergodic decompositions for folding and unfolding paths in Outer space

Abstract

We relate ergodic-theoretic properties of a very small tree or lamination to the behavior of folding and unfolding paths in Outer space that approximate it, and we obtain a criterion for unique ergodicity in both cases. Our main result is that non-unique ergodicity gives rise to a transverse decomposition of the folding/unfolding path. It follows that non-unique ergodicity leads to distortion when projecting to the complex of free factors, and we give two applications of this fact. First, we show that if a subgroup $H$ of $Out(\FN)$ quasi-isometrically embeds into the complex of free factors via the orbit map, then the limit set of $H$ in the boundary of Outer space consists of trees that are uniquely ergodic and have uniquely ergodic dual lamination. Second, we describe the Poisson boundary for random walks coming from distributions with finite first moment with respect to the word metric on $Out(\FN)$: almost every sample path converges to a tree that is uniquely ergodic and that has a uniquely ergodic dual lamination, and the corresponding hitting measure on the boundary of Outer space is the Poisson boundary. This improves a recent result of Horbez. We also obtain sublinear tracking of sample paths with Lipschitz geodesic rays.