Geodesic
Dynamically ordered energy function for Morse--Smale diffeomorphisms on $3$-manifolds
Informatics and Automation, Differential equations and dynamical systems, Tome 278 (2012), pp. 34-48

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This paper deals with arbitrary Morse–Smale diffeomorphisms in dimension $3$ and extends ideas from the authors' previous studies where the gradient-like case was considered. We introduce a kind of Morse–Lyapunov function, called dynamically ordered, which fits well the dynamics of a diffeomorphism. The paper is devoted to finding conditions for the existence of such an energy function, that is, a function whose set of critical points coincides with the nonwandering set of the considered diffeomorphism. We show that necessary and sufficient conditions for the existence of a dynamically ordered energy function reduce to the type of the embedding of one-dimensional attractors and repellers, each of which is a union of zero- and one-dimensional unstable (stable) manifolds of periodic orbits of a given Morse–Smale diffeomorphism on a closed $3$-manifold. 
@article{TRSPY_2012_278_a3,
     author = {V. Z. Grines and F. Laudenbach and O. V. Pochinka},
     title = {Dynamically ordered energy function for {Morse--Smale} diffeomorphisms on $3$-manifolds},
     journal = {Informatics and Automation},
     pages = {34--48},
     publisher = {mathdoc},
     volume = {278},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2012_278_a3/}
}
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V. Z. Grines; F. Laudenbach; O. V. Pochinka. Dynamically ordered energy function for Morse--Smale diffeomorphisms on $3$-manifolds. Informatics and Automation, Differential equations and dynamical systems, Tome 278 (2012), pp. 34-48. http://geodesic.mathdoc.fr/item/TRSPY_2012_278_a3/