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

This paper deals with arbitrary Morse–Smale diffeomorphisms in dimension $3$ and extends ideas from the authors' previous studies where the gradient-like case was considered. We introduce a kind of Morse–Lyapunov function, called dynamically ordered, which fits well the dynamics of a diffeomorphism. The paper is devoted to finding conditions for the existence of such an energy function, that is, a function whose set of critical points coincides with the nonwandering set of the considered diffeomorphism. We show that necessary and sufficient conditions for the existence of a dynamically ordered energy function reduce to the type of the embedding of one-dimensional attractors and repellers, each of which is a union of zero- and one-dimensional unstable (stable) manifolds of periodic orbits of a given Morse–Smale diffeomorphism on a closed $3$-manifold.