Rigidity of the dynamics of $\mathrm{A}\mathrm{u}\mathrm{t}({\mathsf{F}}_n)$ on representations into a compact group

Let $G$ be a compact Lie group. Let ${\mathsf{F}}_n$ be the free group of rank $n$. We describe the orbits of $\mathsf{A}\mathsf{u}\mathsf{t}({\mathsf{F}}_n)$ on $\mathsf{H}\mathsf{o}\mathsf{m}({\mathsf{F}}_n;G)$ when $n$ is sufficiently large. The dynamics stabilizes: orbit closures and invariant probability measures are algebraic, as in Ratner’s theorems.