Geodesic
Classification of Morse--Smale diffeomorphisms with one-dimensional set of unstable separatrices
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 270 (2010), pp. 62-85.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $M^n$ be a closed orientable manifold of dimension $n>3$. We study the class $G_1(M^n)$ of orientation-preserving Morse–Smale diffeomorphisms of $M^n$ such that the set of unstable separatrices of any $f\in G_1(M^n)$ is one-dimensional and does not contain heteroclinic intersections. We prove that the Peixoto graph (equipped with an automorphism) is a complete topological invariant for diffeomorphisms of class $G_1(M^n)$, and construct a standard representative for any class of topologically conjugate diffeomorphisms. 
@article{TM_2010_270_a4,
     author = {V. Z. Grines and E. Ya. Gurevich and V. S. Medvedev},
     title = {Classification of {Morse--Smale} diffeomorphisms with one-dimensional set of unstable separatrices},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {62--85},
     publisher = {mathdoc},
     volume = {270},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2010_270_a4/}
}
TY  - JOUR
AU  - V. Z. Grines
AU  - E. Ya. Gurevich
AU  - V. S. Medvedev
TI  - Classification of Morse--Smale diffeomorphisms with one-dimensional set of unstable separatrices
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2010
SP  - 62
EP  - 85
VL  - 270
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2010_270_a4/
LA  - ru
ID  - TM_2010_270_a4
ER  - 
%0 Journal Article
%A V. Z. Grines
%A E. Ya. Gurevich
%A V. S. Medvedev
%T Classification of Morse--Smale diffeomorphisms with one-dimensional set of unstable separatrices
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2010
%P 62-85
%V 270
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2010_270_a4/
%G ru
%F TM_2010_270_a4
V. Z. Grines; E. Ya. Gurevich; V. S. Medvedev. Classification of Morse--Smale diffeomorphisms with one-dimensional set of unstable separatrices. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 270 (2010), pp. 62-85. http://geodesic.mathdoc.fr/item/TM_2010_270_a4/

[1] Barden D., The structure of manifolds, Thesis, Cambridge Univ., 1963

[2] Bass H., Heller A., Swan R.G., “The Whitehead group of a polynomial extension”, Publ. Math. IHES, 22, 1964, 61–79 | DOI | MR

[3] Bezdenezhnykh A.N., Grines V.Z., “Dynamical properties and topological classification of gradient-like diffeomorphisms on two-dimensional manifolds. I”, Sel. Math. Sov., 11:1 (1992), 1–11 | MR | MR | Zbl

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

[5] Bonatti C., Grines V., Medvedev V., Pécou E., “Topological classification of gradient-like diffeomorphisms on 3-manifolds”, Topology, 43 (2004), 369–391 | DOI | MR | Zbl

[6] Bonatti C., Grines V., Pécou E., “Two-dimensional links and diffeomorphisms on 3-manifolds”, Ergodic Theory and Dyn. Syst., 22 (2002), 687–710 | DOI | MR | Zbl

[7] Bonatti Kh., Grines V.Z., Pochinka O.V., “Klassifikatsiya diffeomorfizmov Morsa–Smeila s konechnym mnozhestvom geteroklinicheskikh orbit na 3-mnogoobraziyakh”, DAN, 396:4 (2004), 439–442 | MR | Zbl

[8] Brown M., “Locally flat imbeddings of topological manifolds”, Ann. Math. Ser. 2, 75:2 (1962), 331–341 | DOI | MR | Zbl

[9] Bryant J.L., Seebeck C.L. III, “Locally nice embeddings in codimension three”, Quart. J. Math. Oxford, 21 (1970), 265–272 | DOI | MR | Zbl

[10] Cappell S.E., Shaneson J.L., “On 4-dimensional s-cobordisms”, J. Diff. Geom., 22:1 (1985), 97–115 | MR | Zbl

[11] Chapman T.A., “Topological invariance of Whitehead torsion”, Amer. J. Math., 96 (1974), 488–497 | DOI | MR | Zbl

[12] Donaldson S.K., “Irrationality and the h-cobordism conjecture”, J. Diff. Geom., 26:1 (1987), 141–168 | MR | Zbl

[13] Freedman M.H., Teichner P., “4-Manifold topology. I: Subexponential groups”, Invent. math., 122:3 (1995), 509–529 | DOI | MR | Zbl

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

[15] Grines V.Z., Gurevich E.Ya., “O diffeomorfizmakh Morsa–Smeila na mnogoobraziyakh razmernosti bolshe trekh”, DAN, 416:1 (2007), 15–17 | MR | Zbl

[16] Grines V.Z., Gurevich E.Ya., Medvedev V.S., “Graf Peikshoto diffeomorfizmov Morsa–Smeila na mnogoobraziyakh razmernosti, bolshei trekh”, Tr. MIAN, 261, 2008, 61–86 | MR | Zbl

[17] 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

[18] Higman G., “The units of group-rings”, Proc. London Math. Soc., 46 (1940), 231–248 | DOI | MR

[19] Khirsh M., Differentsialnaya topologiya, Mir, M., 1979 | MR

[20] Hudson J.F.P., Zeeman E.C., “On combinatorial isotopy”, Publ. Math. IHES, 19, 1964, 69–94 | DOI | MR | Zbl

[21] Keldysh L.V., Topologicheskie vlozheniya v evklidovo prostranstvo, Tr. MIAN, 81, Nauka, M., 1966

[22] Kirby R.C., Siebenmann L.C., Foundational essays on topological manifolds, smoothings, and triangulations, Ann. Math. Stud., 88, Princeton Univ. Press, Princeton (NJ), 1977 | MR | Zbl

[23] Kosnevski Ch., Nachalnyi kurs algebraicheskoi topologii, Mir, M., 1983 | MR

[24] Kwasik S., “On low dimensional S-cobordisms”, Comment. Math. Helv., 61 (1986), 415–428 | DOI | MR | Zbl

[25] Mazur B. Differential topology from the point of view of simple homotopy theory, Publ. Math. IHES, 15, Inst. Hautes Étud. Sci., Bures-sur-Yvette, 1963 | MR | Zbl

[26] Milnor J.W., Lectures on the h-cobordism theorem, Princeton Univ. Press, Princeton (NJ), 1965 ; Milnor Dzh., Teorema ob h-kobordizme, Mir, M., 1969 | MR | MR

[27] Palis Zh., di Melu V., Geometricheskaya teoriya dinamicheskikh sistem: Vvedenie, Mir, M., 1986 | MR

[28] Palis J., Smale S., “Structural stability theorems”, Global analysis, Amer. Math. Soc., Providence (RI), 1970, 223–231 ; Pali Dzh., Smeil S., “Teoremy strukturnoi ustoichivosti”, Matematika, 13, no. 2, 1969, 145–155 | DOI | MR

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

[30] Rokhlin V.A., Fuks D.B., Nachalnyi kurs topologii: Geometricheskie glavy, Nauka, M., 1977 | MR | Zbl

[31] Rurk K., Sanderson B., Vvedenie v kusochno lineinuyu topologiyu, Mir, M., 1974 | MR

[32] Shilnikov L.P., Shilnikov A.L., Turaev D.V., Chua L., Metody kachestvennoi teorii v nelineinoi dinamike, Ch. 1, In-t kompyut. issled., Moskva; Izhevsk, 2004

[33] Smale S., “Differentiable dynamical systems”, Bull. Amer. Math. Soc., 73:6 (1967), 747–817 ; S. Smeil, “Differentsiruemye dinamicheskie sistemy”, UMN, 25:1 (1970), 113–185 | DOI | MR | Zbl | MR

[34] Spener E., Algebraicheskaya topologiya, Mir, M., 1971 | MR

[35] Stallings J.R., “On infinite processes leading to differentiability in the complement of a point”, Differential and combinatorial topology: A symposium in honor of M. Morse, Princeton Univ. Press, Princeton (NJ), 1965, 245–254 | MR | Zbl

[36] Stallings J.R., Lectures on polyhedral topology, Tata Inst. Fund. Res., Bombay, 1968 | MR | Zbl

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

[38] Whitehead J.H.C., “Simple homotopy types”, Amer. J. Math., 72 (1950), 1–57 | DOI | MR | Zbl