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 
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/
@article{TRSPY_2012_278_a3,
     author = {V. Z. Grines and F. Laudenbach and O. V. Pochinka},
     title = {Dynamically ordered energy function for {Morse{\textendash}Smale} diffeomorphisms on $3$-manifolds},
     journal = {Informatics and Automation},
     pages = {34--48},
     year = {2012},
     volume = {278},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2012_278_a3/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Bonatti C., Grines V., Medvedev V., Pécou E., “Three-manifolds admitting Morse–Smale diffeomorphisms without heteroclinic curves”, Topology Appl., 117 (2002), 335–344 | DOI | MR | Zbl

[2] Bonatti Kh., Grines V.Z., Pochinka O.V., “Klassifikatsiya diffeomorfizmov Morsa–Smeila s konechnym mnozhestvom geteroklinicheskikh orbit na 3-mnogoobraziyakh”, Tr. MIAN, 250, 2005, 5–53 | MR | Zbl

[3] Grines V.Z., Laudenbakh F., Pochinka O.V., “Energeticheskaya funktsiya dlya gradientnopodobnykh diffeomorfizmov na 3-mnogoobraziyakh”, DAN, 422:3 (2008), 299–301 | MR | Zbl

[4] Grines V., Laudenbach F., Pochinka O., “Self-indexing energy function for Morse–Smale diffeomorphisms on 3-manifolds”, Moscow Math. J., 9:4 (2009), 801–821 | MR | Zbl

[5] Grines V.Z., Zhuzhoma E.V., Medvedev V.S., “Novye sootnosheniya dlya sistem Morsa–Smeila s trivialno vlozhennymi odnomernymi separatrisami”, Mat. sb., 194:7 (2003), 25–56 | DOI | MR | Zbl

[6] Milnor Dzh., Teoriya Morsa, Mir, M., 1965 | MR

[7] Palis J., “On Morse–Smale dynamical systems”, Topology, 8 (1969), 385–404 | DOI | MR

[8] Pixton D., “Wild unstable manifolds”, Topology, 16 (1977), 167–172 | DOI | MR | Zbl

[9] Robinson C., Dynamical systems: Stability, symbolic dynamics, and chaos, Stud. Adv. Math., 2nd ed., CRC Press, Boca Raton, FL, 1999 | MR | Zbl

[10] Smeil S., “Differentsiruemye dinamicheskie sistemy”, UMN, 25:1 (1970), 113–185 | MR

[11] Smale S., “On gradient dynamical systems”, Ann. Math. Ser. 2, 74 (1961), 199–206 | DOI | MR | Zbl

[12] Waldhausen F., “On irreducible 3-manifolds which are sufficiently large”, Ann. Math. Ser. 2, 87 (1968), 56–88 | DOI | MR | Zbl