Geodesic
On Morse–Smale Diffeomorphisms without Heteroclinic Intersections on Three-Manifolds
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 236 (2002), pp. 66-78 
Ch. Bonatti; V. Z. Grines; V. S. Medvedev; E. Peku. On Morse–Smale Diffeomorphisms without Heteroclinic Intersections on Three-Manifolds. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 236 (2002), pp. 66-78. http://geodesic.mathdoc.fr/item/TM_2002_236_a7/
@article{TM_2002_236_a7,
     author = {Ch. Bonatti and V. Z. Grines and V. S. Medvedev and E. Peku},
     title = {On {Morse{\textendash}Smale} {Diffeomorphisms} without {Heteroclinic} {Intersections} on {Three-Manifolds}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {66--78},
     year = {2002},
     volume = {236},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2002_236_a7/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

A class of Morse–Smale diffeomorphisms is considered that do not admit heteroclinic intersections and are defined on three-manifolds. To each diffeomorphism $f$, we associate an enriched graph $G(f)$ and, for each sink $\omega$, we define a scheme $S(\omega )$ which is a link of tori, the Klein bottle, and simple closed curves embedded in $S^2\times S^1$. We show that diffeomorphisms $f_1$ and $f_2$ are topologically conjugate if and only if (1) the corresponding graphs $G(f_1)$ and $G(f_2)$ are isomorphic and the permutations induced by the dynamics $f_1$ and $f_2$ on the vertices and edges of the graphs are conjugate; (2) two sinks corresponding to isomorphic vertices have equivalent schemes; and (3) for any two saddles corresponding to isomorphic vertices and having one-dimensional unstable manifolds, the corresponding pairs of curves in $S^2\times S^1$ associated with the one-dimensional separatrices are concordantly embedded.

[1] Bezdenezhnykh A. N., Grines V. Z., “Dinamicheskie svoistva i topologicheskaya klassifikatsiya gradientno-podobnykh diffeomorfizmov na dvumernykh mnogoobraziyakh, I, II”, Metody kachestvennoi teorii differentsialnykh uravnenii, Gork. gos. un-t, Gorkii, 1984, 22–38 ; 1987, 24–31 ; Sel. Math. Sov., 11:1 (1992), 1–11, 13–17 | MR | Zbl | MR | Zbl | MR

[2] Bezdenezhnykh A. N., Grines V. Z., “Realization of gradient-like diffeomorphisms of two-dimensional manifolds”, Sel. Math. Sov., 11:1 (1992), 19–23 | MR

[3] Bonatti Ch., Grines V., “Knots as topological invariant for gradient-like diffeomorphisms of the sphere $S^3$”, J. Dyn. and Contr. Syst., 6:4 (2000), 579–602 | DOI | MR | Zbl

[4] Bonatti Ch., Langevin R., Diffeomorphismes de Smale des surfaces, Astérisque, 250, Soc. Math. France, Paris, 1998, 235 pp. | MR | Zbl

[5] Bonatti Ch., Grines V., Medvedev V., Pécou E., Three-manifolds admitting Morse–Smale diffeomorphisms without heteroclinic curves, Prepubl. Rech. No 214, Univ. Bourgogne, 2000, 8 pp.; Topol. and Appl. (to appear)

[6] Bonatti Ch., Grines V., Pécou E., Bidimensional links and diffeomorphisms on 3-manifolds, Prepubl. Rech. No 225, Univ. Bourgogne, 2000, 21 pp.; Ergod. Th. and Dyn. Syst. (to appear)

[7] Grines V. Z., “Topologicheskaya klassifikatsiya diffeomorfizmov Morsa–Smeila s konechnym mnozhestvom geteroklinicheskikh traektorii na poverkhnostyakh”, Mat. zametki., 54:3 (1993), 3–17 | MR | Zbl

[8] Grines V. Z., Medvedev V. S., “O topologicheskoi sopryazhennosti trekhmernykh gradientnopodobnykh diffeomorfizmov s trivialno vlozhennym mnozhestvom separatris sedlovykh nepodvizhnykh tochek”, Mat. zametki, 66:6 (1999), 945–948 | MR | Zbl

[9] Epstein D. B. A., “Curves on 2-manifolds and isotopies”, Acta Math., 115 (1966), 83–107 | DOI | MR | Zbl

[10] Langevin R., “Quelques nouveaux invariants des difféomorphismes Morse–Smale d'une surface”, Ann. Inst. Fourier (Grenoble), 43:1 (1993), 265–278 | MR | Zbl

[11] Smale S., “Differentiable dynamical systems”, Bull. Amer. Math. Soc., 73:6 (1967), 747–817 | DOI | MR

[12] Umanskii Ya. L., “Neobkhodimye i dostatochnye usloviya topologicheskoi ekvivalentnosti trekhmernykh sistem Morsa–Smeila s konechnym mnozhestvom osobykh traektorii”, Mat. sb., 181:2 (1990), 212–239