Criterion for the Existence of an Energy Function for a Regular Homeomorphism of the 3-Sphere

C. Conley's fundamental theorem of the theory of dynamical systems states that every dynamical system, even a nonsmooth one (i.e., a continuous flow or a discrete dynamical system generated by a homeomorphism), admits a continuous Lyapunov function. A Lyapunov function is strictly decreasing along the trajectories of the dynamical system outside the chain recurrent set and is constant on the chain component. A Lyapunov function whose set of critical points coincides with the chain recurrent set of the dynamical system is called an energy function; it has the closest relationship with the dynamics. However, not every dynamical system has an energy function. In particular, according to D. Pixton, even a structurally stable diffeomorphism with nonwandering set consisting of four fixed points may not have a smooth energy function. Our main result in this paper is a criterion for the existence of a continuous Morse energy function for regular homeomorphisms of the $3$-sphere, according to which the existence of such a function is equivalent to the asymptotic triviality of one-dimensional saddle manifolds. The criterion generalizes the results of V. Z. Grines, F. Laudenbach, and O. V. Pochinka for Morse–Smale $3$-diffeomorphisms in the case when the ambient manifold is the three-dimensional sphere. In particular, our criterion implies that Pixton's examples do not admit even a continuous energy function.