Applications of weak attraction theory in Out($F_n$)

Given a free group of rank r >= 3 and two exponentially growing outer automorphisms {\psi} and {\phi} with dual lamination pairs {\Lambda^\pm}_{\psi} and {\Lambda^\pm}_{\phi} associated to them, which satisfy a notion of independence described in this paper, we will use the pingpong techniques developed in recent works of Handel and Mosher to show that there exists an integer M > 0, such that for every m, n >= M, {\psi}^m and {\phi}^n will freely generate a group of rank two and every element of this free group which is not conjugate to a power of the generators will be fully irreducible and hyperbolic. The main corollary is a complete understanding of composition of powers of fully irreducible elements. As bonus we will reprove a theorem in MCG of surfaces with one boundary component and also have a new proof of a theorem of Kapovich and Lustig. The final section also has a relativised version of the main theorem which was recently used my Handel and Mosher to prove the second bounded cohomology alternative for Out($F_n$) in arXiv:1511.06913