Structural stability of attractor–repellor endomorphisms with singularities

Abstract

AbstractWe prove a theorem on the 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 an expanding compact subset with a hyperbolic attractor on which f acts bijectively. The statement of this result is both infinitesimal and dynamical. To our knowledge, this is the first in this hybrid direction. Our results also generalize Mather’s theorem in singularity theory, which states that infinitesimal stability implies structural stability for composed mappings to the larger category of laminations.