A train track directed random walk on Out(Fr)

Several known results, by Rivin, Calegari-Maher and Sisto, show that an element φn ∈ Out (Fr), obtained after n steps of a simple random walk on Out (Fr), is fully irreducible with probability tending to 1 as n → ∞. In this paper, we construct a natural "train track directed" random walk 𝒲 on Out (Fr) (where r ≥ 3). We show that, for the element φn ∈ Out (Fr), obtained after n steps of this random walk, with asymptotically positive probability the element φn has the following properties: φn is an ageometric fully irreducible, which admits a train track representative with no periodic Nielsen paths and exactly one nondegenerate illegal turn, that φn has "rotationless index" [Formula: see text] (so that the geometric index of the attracting tree Tφn of φn is 2r - 3), has index list [Formula: see text] and the ideal Whitehead graph being the complete graph on 2r - 1 vertices, and that the axis bundle of φn in the Outer space CV r consists of a single axis.