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.

Stability in Outer space

“…As a consequence of Proposition 3.2, Corollary 4.1 and Theorem 1.1 of [NPR14] we can now state that indeed, a contracting line of minima γ has two endpoints in CV(F n ). This means that if γ is such a line of minima, then [γ(t)] ⊂ CV(F n ) converges as t → ∞ to an endpoint tree in ∂CV(F n ).…”

“…We call [T ] the endpoint tree of the path. The tree is called uniquely ergometric (see [NPR14] for this notion) if it admits a unique non-atomic length measure up to scale. Here a length measure assigns to each non-degenerate segment of T a positive length, and this length is invariant under the action of F n and additive with respect to concatenation.…”

“…The sequence is stabilized if for large enough i the restriction of f ni,nj to E ni is a permutation. The reasoning in Example 6.2 shows that fast folding paths containing stabilized proper subgraphs are not stable and hence we may assume from now on that γ is reduced, ie it does not contain stabilized subgraphs (compare [NPR14]).…”

“…We now argue by contradiction and we assume that the endpoint tree T of γ is not uniquely ergometric. Since by the above remark the sequence G ni is reduced, we can apply the results from Section 6 of [NPR14].…”

“…However, their arguments are very different from the arguments we use, and their main goal is different in flavor as well. There is also some overlap with the work [NPR14] whose main result is a key ingredient in the proof of Theorem 2.…”