Quasi-Energy Function for Morse–Smale 3-Diffeomorphisms with Fixed Points with Pairwise Distinct Indices

Abstract

The present paper is devoted to a lower bound for the number of critical points of the Lyapunov function for Morse–Smale 3-diffeomorphisms with fixed points with pairwise distinct indices. It is known that, in the presence of a single noncompact heteroclinic curve, the supporting manifold of the diffeomorphisms under consideration is a 3-sphere, and the class of topological conjugacy of such a diffeomorphism \(f\) is completely determined by the equivalence class (there exist infinitely many of them) of the Hopf knot \(L_{f}\), which is a knot in the generating class of the fundamental group of the manifold \(\mathbb S^2\times \mathbb S^1\).

Moreover, any Hopf knot is realized by some diffeomorphism of the class under consideration. It is known that the diffeomorphisms defined by the standard Hopf knot \(L_0=\{s\}\times \mathbb S^1\) have an energy function, which is a Lyapunov function whose set of critical points coincides with the chain recurrent set. However, the set of critical points of any Lyapunov function of a diffeomorphism \(f\) with a nonstandard Hopf knot is strictly greater than the chain recurrent set of the diffeomorphism.

In the present paper, for the diffeomorphisms defined by generalized Mazur knots, a quasi-energy function has been constructed, which is a Lyapunov function with a minimum number of critical points.