A quasi-energy function for Pixton diffeomorphisms defined by generalized Mazur knots

TL;DR

This paper constructs a quasi-energy function for Pixton diffeomorphisms associated with generalized Mazur knots, providing a lower estimate for the critical points of Lyapunov functions in this context.

Contribution

It introduces a quasi-energy function for Pixton diffeomorphisms defined by generalized Mazur knots, extending previous results to a broader class of knots.

Findings

Constructed a quasi-energy function for Pixton diffeomorphisms with generalized Mazur knots.

Provided a lower estimate for the number of critical points of Lyapunov functions.

Extended the class of known Pixton diffeomorphisms with explicit energy functions.

Abstract

In this paper we give a lower estimate for the number of critical points of the Lyapunov function for Pixton diffeomorphisms (i.e. Morse-Smale diffeomorphisms in dimension 3 whose chain recurrent set consists of four points: one source, one saddle and two sinks). Ch. Bonatti and V. Grines proved that the class of topological equivalence of such diffeomorphism $f$ is completely defined by the equivalency class of the Hopf knot $L_{f}$ that is the knot in the generating class of the fundamental group of the manifold $\mathbb S^2\times\mathbb S^1$. They also proved that there are infinitely many such classes and that any Hopf knot can be realized by a Pixton diffeomorphism. D.~Pixton proved that diffeomorphisms defined by the standard Hopf knot $L_0=\{s\}\times \mathbb S^1$ have an energy function (Lyapunov function) whose set of critical points coincide with the chain recurrent set whereas the set of critical points of any Lyapunov function for Pixton diffeomorphism with nontrivial (i.e. non equivalent to the standard) Hopf knot is strictly larger than the chain recurrent set of the diffeomorphism. The Lyapunov function for Pixton diffeomorphism with minimal number of critical points is called the quasi-energy function. In this paper we construct a quasi-energy function for Pixton diffeomorphisms defined by a generalized Mazur knot.