5. The Classification of the Gradient-Like Diffeomorphisms on 3-Manifolds

In the classical papers by S. Smale and J. Palis the proof of structural stability of Morse-Smale diffeomorphisms was based on the construction of a system of tubular neighborhoods. We present the similar construction for gradient-like 3-diffeomorphisms using the idea of representation of the dynamics of the system as “attractor-repeller” and the consideration of the space of the wandering orbits. We come to the compatible system of neighborhoods which plays the key role in the topological classification. Let \(MS_0(M^3)\) denote the class of gradient-like diffeomorphisms on the manifold \(M^3\). In this chapter we give the complete topological classification of the diffeomorphisms of this class by means of the topological invariant called the scheme of the diffeomorphism which generalizes the invariants for the Pixton class. The scheme is a simple 3-manifold whose fundamental group admits an epimorphism to the group \(\mathbb Z\) and a system of tori and Klein bottles smoothly embedded into this manifold. The presented results are for the most part from the paper [5]. In the papers [1, 3, 4, 6‐10, 14] one can find topological classification of some special classes of the Morse–Smale diffeomorphisms on 2-manifolds.