Geodesic
Quasi-Energy Function for Diffeomorphisms with Wild Separatrices
Matematičeskie zametki, Tome 86 (2009) no. 2, pp. 175-183

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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.
Keywords: Morse–Smale diffeomorphism, Lyapunov function, Morse theory, saddle, sink, separatrix, wild embedding, Heegaard splitting, cobordism.
Mots-clés : source 
@article{MZM_2009_86_2_a2,
     author = {V. Z. Grines and F. Laudenbach and O. V. Pochinka},
     title = {Quasi-Energy {Function} for {Diffeomorphisms} with {Wild} {Separatrices}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {175--183},
     publisher = {mathdoc},
     volume = {86},
     number = {2},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2009_86_2_a2/}
}
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V. Z. Grines; F. Laudenbach; O. V. Pochinka. Quasi-Energy Function for Diffeomorphisms with Wild Separatrices. Matematičeskie zametki, Tome 86 (2009) no. 2, pp. 175-183. http://geodesic.mathdoc.fr/item/MZM_2009_86_2_a2/