Quasi-energy function for diffeomorphisms with wild separatrices

We consider the class G4 of Morse—Smale diffeomorphisms on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{S} $$\end{document}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 G4,1 ⊂ G4 of diffeomorphisms inducing a special Heegaard splitting of genus 1 of the sphere \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{S} $$\end{document}3. For each diffeomorphism in G4,1, we present a quasi-energy function with six critical points.