Quasi-Energy Function for Diffeomorphisms with Wild Separatrices

We consider the class $G_4$ of Morse–Smale diffeomorphisms on $\mathbb S^3$ with nonwandering set consisting of four fixed points (namely, one saddle, two sinks, and one source). According to Pixton, this class contains a diffeomorphism that does not have an energy function, i.e., a Lyapunov function whose set of critical points coincides with the set of periodic points of the diffeomorphism itself. We define a quasi-energy function for any Morse–Smale diffeomorphism as a Lyapunov function with the least number of critical points. Next, we single out the class $G_{4,1}\subset G_4$ of diffeomorphisms inducing a special Heegaard splitting of genus 1 of the sphere $\mathbb S^3$. For each diffeomorphism in $G_{4,1}$, we present a quasi-energy function with six critical points.