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

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 is completely determined by the equivalence class (there exist infinitely many of them) of the Hopf knot , which is a knot in the generating class of the fundamental group of the manifold . 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 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 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. [ABSTRACT FROM AUTHOR]