Structural stability of attractor-repellor endomorphisms with singularities

TL;DR

This paper establishes a new theorem on the structural stability of smooth attractor-repellor endomorphisms with singularities on compact manifolds, extending classical results to a hybrid dynamical and singularity context.

Contribution

It introduces the first structural stability theorem for attractor-repellor endomorphisms with singularities, generalizing Mather's theorem to laminations.

Findings

Proves structural stability for a class of endomorphisms with singularities.

Extends Mather's theorem from mappings to laminations.

Provides a hybrid infinitesimal and dynamical stability result.

Abstract

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.