Quasi-energy function for diffeomorphisms with wild separatrices

Abstract

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 ⊂ 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.