  We show that the level sets of automorphisms of free groups with respect to
the Lipschitz metric are connected as subsets of Culler-Vogtmann space. In fact
we prove our result in a more general setting of deformation spaces. As
applications, we give metric solutions of the conjugacy problem for irreducible
automorphisms and the detection of reducibility. We additionally prove
technical results that may be of independent interest --- such as the fact that
the set of displacements is well ordered.
