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 $k\ge 5$ there does not exist a continuous map $\partial CV(F_k)\to\mathbb PCurr(F_k)$thatiseither$Out(F_k)$-equivariantor$Out(F_k)$-anti-equivariant\mathrm{.}Here$\partial CV(F_k)$ is the “length-function” boundary of Culler-Vogtmann’s Outer space $CV(F_k)$, and $\mathbb{P}Curr(F_k)$ is the space of projectivized geodesic currents for $F_k$. We also prove that, if $k\ge 3$, for the action of $Out(F_k)$ on $\mathbb{P}Curr(F_k)$ and for the diagonal action of $Out(F_k)$ on the product space $\partial CV(F_k)\times \mathbb PCurr(F_k)$thereexistuniquenon-emptyminimalclosed$Out(F_k)$-invariantsets\mathrm{.}Ourresultsimplythatfor$k\ge 3$anycontinuous$Out(F_k)$-equivariantembeddingof$CV(F_k)$into$\mathbb PCurr(F_k)$ (such as the Patterson-Sullivan embedding) produces a new compactification of Outer space, different from the usual “length-function” compactification $\overline{CV(F_k)}=CV(F_k)\cup \mathrm{\partial }CV(F_k)$.