Dynamically ordered energy function for Morse-Smale diffeomorphisms on 3-manifolds

TL;DR

This paper introduces a new type of Morse-Lyapunov function called dynamically ordered, for Morse-Smale diffeomorphisms on 3-manifolds, linking its existence to the embedding of attractors and repellers.

Contribution

It extends previous work to 3-dimensional cases and establishes conditions for the existence of a dynamically ordered energy function based on embedding properties.

Findings

Necessary and sufficient conditions for the energy function existence.

Reduction of conditions to embedding types of attractors and repellers.

Extension of gradient-like case to general Morse-Smale diffeomorphisms.

Abstract

This note deals with arbitrary Morse-Smale diffeomorphisms in dimension 3 and extends ideas from \cite{GrLaPo}, \cite{GrLaPo1}, where gradient-like case was considered. We introduce a kind of Morse-Lyapunov function, called dynamically ordered, which fits well dynamics of diffeomorphism. The paper is devoted to finding conditions to the existence of such an energy function, that is, a function whose set of critical points coincides with the non-wandering set of the considered diffeomorphism. We show that the necessary and sufficient conditions to the existence of a dynamically ordered energy function reduces to the type of embedding of one-dimensional attractors and repellers of a given Morse-Smale diffeomorphism on a closed 3-manifold.