Self-indexing energy function for Morse-Smale diffeomorphisms on 3-manifolds

TL;DR

This paper investigates conditions under which Morse-Smale diffeomorphisms on 3-manifolds admit a self-indexing energy function, linking it to the embedding of stable and unstable manifolds and a special type of Heegaard splitting.

Contribution

It establishes the equivalence between the existence of a self-indexing energy function and a specific Heegaard splitting for Morse-Smale diffeomorphisms on 3-manifolds.

Findings

Conditions involving stable and unstable manifold embeddings are necessary for the energy function.

Existence of a self-indexing energy function is equivalent to a special Heegaard splitting.

Provides criteria for the existence of energy functions in 3-dimensional Morse-Smale systems.

Abstract

The paper is devoted to finding conditions to the existence of a self-indexing energy function for Morse-Smale diffeomorphisms on a 3-manifold. These conditions involve how the stable and unstable manifolds of saddle points are embedded in the ambient manifold. We also show that the existence of a self-indexing energy function is equivalent to the existence of a Heegaard splitting of a special type with respect to the considered diffeomorphism.