The actions of Out(F_k) on the boundary of Outer space and on the space of currents: Minimal sets and equivariant incompatibility

We prove that for k > 5 there does not exist a continuous map ∂CV(F_k) → PCurr(F_k) that is either Out(F _k)-equivariant or Out(F_k)-anti-equivariant. Here ∂CV(F_k) is the 'length function' boundary of Culler-Vogtmann's Outer space CV(F_k), and PCurr(F_k) is the space of projectivized geodesic currents for F_k. We also prove that, if k ≥ 3, for the action of Out(F_k) on ℙCwrr(F_k) and for the diagonal action of Out(F_k) on the product space ∂CV(F _k) × PCurr(F_k), there exist unique non-empty minimal closed Out(F_k)-invariant sets. Our results imply that for k ≥ 3 any continuous Out(F_k)-equivariant embedding of CV(F_k) into ℙCurr(F_k) (such as the Patterson-Sullivan embedding) produces a new compactification of Outer space, different from the usual 'length function' compactification CV(F_k) = CV(F_k) ∪ ∂CV(F_k).