  In this article, we associate to isometries of CAT(0) cube complexes specific
subspaces, referred to as \emph{median sets}, which play a similar role as
minimising sets of semisimple isometries in CAT(0) spaces. Various applications
are deduced, including a cubulation of centralisers, a splitting theorem, a
proof that Dehn twists in mapping class groups must be elliptic for every
action on a CAT(0) cube complex, a cubical version of the flat torus theorem,
and a structural theorem about polycyclic groups acting on CAT(0) cube
complexes.
