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

The paper is devoted to finding conditions for the existence of a self-indexing energy function for Morse–Smale diffeomorphisms on a 3-manifold $M^3$. 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 $M^3$ of a special type with respect to the considered diffeomorphism.