  We establish an equivalence between infinitely many asymptotically stable
periodic solutions and subsumed homoclinic connections for $N$-dimensional
piecewise-linear continuous maps. These features arise as a codimension-three
phenomenon. The periodic solutions are single-round: they each involve one
excursion away from a central saddle-type periodic solution. The homoclinic
connection is subsumed in the sense that one branch of the unstable manifold of
the saddle solution is contained entirely within its stable manifold. The
results are proved by using exact expressions for the periodic solutions and
components of the stable and unstable manifolds which are available because the
maps are piecewise-linear. We also describe a practical approach for finding
this phenomenon in the parameter space of a map and illustrate the results with
the three-dimensional border-collision normal form.
