The actions of Out(Fk) 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 a partial derivative CV(F-k) -> PCurr(F-k) that is either Out(F-k)-equivariant or Out(F-k)-anti-equivariant. Here partial derivative 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 PCurr(F-k) and for the diagonal action of Out(F-k) on the product space partial derivative CV(F-k) x 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 PattersonSullivan embedding) produces a new compactification of Outer space, different from the usual 'length function' compactification CV(F-k) = CV (F-k) U partial derivative CV (F-k).