  We prove a theorem on structural stability of smooth attractor-repellor
endomorphisms of compact manifolds, with singularities. By attractor-repellor,
we mean that the non-wandering set of the dynamics $f$ is the disjoint union of
a repulsive compact subset with a hyperbolic attractor on which $f$ acts
bijectively. The statement of this result is both infinitesimal and dynamical.
Up to our knowledge, this is the first in this hybrid direction. Our results
generalize also a Mather's theorem in singularity theory which states that
infinitesimal stability implies structural stability for composed mappings, to
the larger category of laminations.
