Quasi-Energy Function for Diffeomorphisms with Wild Separatrices

We consider the class G(4) of Morse-Smale diffeomorphisms on 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 in energy function, i.e., a Lyanpunov 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 of G(4) of diffeomorphisms inducing a special Heegaard Splitting Of genus 1 of the sphere S(3). For each diffeomorphism in G(4,1), we present a quasi-energy function with six critical points.