Limits of conjugacy classes under iterates of hyperbolic elements of Out(F)

For a free group F of finite rank such that rank ( F) ≥ 3 , we prove that the set of weak limits of a conjugacy class in F under iterates of some hyperbolic φ ∈ Out ( F) is equal to the collection of generic leaves and lines with endpoints in attracting fixed points of φ . As an application we describe the ending lamination set for a hyperbolic extension of F by a hyperbolic element of Out( F ) in a new way and use it to prove results about Cannon–Thurston maps for such extensions. We also use it to derive conditions for quasiconvexity of finitely generated, infinite index subgroups of F in the extension group. These results generalize similar results obtained in [23] and [19] and use different techniques.