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For a free group $\mathbb{F}$ of finite rank such that$\text{rank}(\mathbb{F})\geq 3$, we prove that the set of weak limits of aconjugacy class in $\mathbb{F}$ under iterates of some hyperbolic$\phi\in\mathsf{Out(\mathbb{F})}$ is equal to the collection of generic leavesand singular lines of $\phi$. As an application we describe the endinglamination set for a hyperbolic extension of $\mathbb{F}$ by a hyperbolicsubgroup of $\mathsf{Out(\mathbb{F})}$ in a new way and use it to prove resultsabout Cannon-Thurston maps for such extensions. We also use it to deriveconditions for quasiconvexity of finitely generated, infinite index subgroupsof $\mathbb{F}$ in the extension group. These results generalize similarresults obtained by Mahan Mj, Kapovich-Lustig and use different techniques.

Limits of conjugacy classes under iterates of Hyperbolic elements of $\mathsf{Out(\mathbb{F})}$