On embedding of the Morse--Smale diffeomorphisms in a topological flow

V. Z. Grines ; E. Ya. Gurevich ; O. V. Pochinka

Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 66 (2020) no. 2, pp. 160-181.

Résumé

This review presents the results of recent years on solving of the Palis problem on finding necessary and sufficient conditions for the embedding of Morse–Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse–Smale diffeomorphisms given on manifolds of dimension two. The result for the circle is a trivial exercise. In dimensions three and higher new effects arise related to the possibility of wild embeddings of closures of invariant manifolds of saddle periodic points that leads to additional obstacles for Morse–Smale diffeomorphisms to embed in topological flows. The progress achieved in solving of Palis's problem in dimension three is associated with the recently obtained complete topological classification of Morse–Smale diffeomorphisms on three-dimensional manifolds and the introduction of new invariants describing the embedding of separatrices of saddle periodic points in a supporting manifold. The transition to a higher dimension requires the latest results from the topology of manifolds. The necessary topological information, which plays key roles in the proofs, is also presented in the survey.

Bibliographie

[1] Ch. Bonatti, V. Z. Grines, O. V. Pochinka, “Classification of Morse–Smale diffeomorphisms with finite set of heteroclinic orbits on 3-manifolds”, Rep. Acad. Sci. USSR, 396:4 (2004), 439–442 (in Russian)

[2] 2017, 35–49

[3] Ch. Bonatti, V. Z. Grines, O. V. Pochinka, “Realization of Morse–Smale diffeomorphisms on 3-manifolds”, Proc. Math. Inst. Russ. Acad. Sci., 297, 2017, 46–61 (in Russian)

[4] M. I. Brin, “On embedding of a diffeomorphism in a flow”, Bull. Higher Edu. Inst. Ser. Math., 8 (1972), 19–25 (in Russian)

[5] 2008, 59–83

[6] V. Z. Grines, E. Ya. Gurevich, V. S. Medvedev, “Peixoto graph of Morse–Smale diffeomorphisms on manifolds of dimension greater than three”, Proc. Math. Inst. Russ. Acad. Sci., 261, 2008, 61–86 (in Russian)

[7] 2010, 57–79

[8] V. Z. Grines, E. Ya. Gurevich, V. S. Medvedev, “On topological classification of Morse–Smale diffeomorphisms with one-dimensional set of unstable separatrices on manifolds of dimension greater than 3”, Proc. Math. Inst. Russ. Acad. Sci., 270, 2010, 62–86 (in Russian)

[9] Math. Notes, 91:5 (2012), 742–745

[10] V. Z. Grines, E. Ya. Gurevich, O. V. Pochinka, V. S. Medvedev, “On embedding of Morse–Smale diffeomorphisms in a flow on manifolds of dimension greater than two”, Math. Notes, 91:5 (2012), 791–794 (in Russian)

[11] 103–124

[12] V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev, O. V. Pochinka, “Global attractor and repeller of Morse–Smale diffeomorphisms”, Proc. Math. Inst. Russ. Acad. Sci., 271, 2010, 111–133 (in Russian)

[13] D. M. Grobman, “On homeomorphism of systems of differential equations”, Rep. Acad. Sci. USSR, 128:5 (1959), 880–881 (in Russian)

[14] D. M. Grobman, “Topological classification of neighborhoods of singular points in $n$-dimensional space”, Math. Digest, 56:1 (1962), 77–94 (in Russian)

[15] W. Hurewicz, H. Wallman, Dimension Theory, Russian translation, Izd-vo inostrannoy literatury, M., 1948

[16] E. V. Zhuzhoma, V. S. Medvedev, “Continuous Morse–Smale flows with three equilibrium states”, Math. Digest, 207:5 (2016), 69–92 (in Russian)

[17] S. Yu. Pilyugin, “Phase diagrams defining Morse–Smale systems without periodic orbits on spheres”, Differ. Equ., 14:2 (1978), 245–254 (in Russian)

[18] O. V. Pochinka, V. Z. Grines, E. Ya. Gurevich, V. S. Medvedev, “On embedding a Morse–Smale diffeomorphism on a 3-manifold in a topological flow”, Math. Digest, 203:12 (2012), 81–104 (in Russian)

[19] Artin E., Fox R. H., “Some wild cells and spheres in three-dimensional space”, Ann. Math., 49 (1948), 979–990

[20] Blankinship W. A., “Generalization of a construction of Antoine”, Ann. Math., 2:3 (1951), 276–297

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

[22] Bonatti C., Grines V., Laudenbach F., Pochinka O., “Topological classification of Morse–Smale diffeomorphisms without heteroclinic curves on 3-manifolds”, Ergodic Theory Dynam. Systems, 39:9 (2019), 2403–2432

[23] Bonatti Ch., Grines V., Medvedev V., Pécou E., “Topological classification of gradient-like diffeomorphisms on $ 3 $-manifolds”, Topology, 43 (2004), 369–391

[24] Bonatti C., Grines V., Pochinka O., “Topological classification of Morse–Smale diffeomorphisms on 3-manifolds”, Duke Math. J., 168:13 (2019), 2507–2558

[25] Brown M., “Locally flat imbeddings of topological manifolds”, Ann. Math. (2), 75:2 (1962), 331–341

[26] Cantrell J. C., “Almost locally flat embeddings of $S^{n-1}$ in $S^{n}$”, Bull. Am. Math. Soc., 69 (1963), 716–718

[27] Cantrell J. C., “Almost locally poliedral curves in Euclidean $n$-space”, Trans. Am. Math. Soc., 107:3 (1963), 451–457

[28] Cantrell J. C., “$n$-frames in Euclidean $k$-space”, Proc. Am. Math. Soc., 15:4 (1964), 574–578

[29] Chernavskii A. V., “Piecewise linear approximation of imbeddings of manifolds in codimensions greater than two”, Sb. Math., 11:3 (1970), 465–466

[30] Daverman R. J., “Embeddings of $(n-1)$-spheres in Euclidean $n$-space”, Bull. Am. Math. Soc., 84:3 (1978), 377–405

[31] Debruner H., Fox R., “A mildly wild embedding of an $n$-frame”, Duke Math. J., 27:3 (1960), 425–429

[32] Dugundji J., Antosiewicz H. A., “Parallelizable flows and Lyapunov's second method”, Ann. Math., 2:73 (1961), 543–555

[33] Foland N. E., Utz W. R., “The embedding of discrete flows in continuous flows”, Ergodic theory, Proc. Int. Symp. (Tulane University, New Orleans, USA, October 1961), Academic Press, New York, 1963, 121–134

[34] Garay B. M., “Discretization and some qualitative properties of ordinary differential equations about equilibria”, Acta Math. Univ. Comenian. (N.S.), 62:2 (1993), 249–275

[35] Garay B. M., “On structural stability of ordinary differential equations with respect to discretization methods”, Numer. Math., 72:4 (1996), 449–479

[36] Grines V., Gurevich E., Pochinka O., “Topological classification of Morse–Smale diffeomorphisms without heteroclinic intersections”, J. Math. Sci. (N.Y.), 208:1 (2015), 81–90

[37] Grines V., Gurevich E., Pochinka O., “On embedding of multidimensional Morse–Smale diffeomorphisms in topological flows”, Mosc. Math. J., 19:4 (2019), 739–760

[38] Grines V., Gurevich E., Pochinka O., On topological classification of Morse–Smale diffeomorphisms on the sphere $S^n$, 2019, arXiv: 1911.10234v2 [math.DS]

[39] Hartman P., “On the local linearization of differential equations”, Proc. Am. Math. Soc., 14:4 (1963), 568–573

[40] Hirsch M., Pugh C., Shub M., Invariant Manifolds, Springer-Verlag, Berlin–Heidelberg–New York, 1977

[41] Hudson J. F., “Concordance and isotopy of PL embeddings”, Bull. Am. Math. Soc., 72:3 (1966), 534–535

[42] Hudson J. F., Zeeman E. C., “On combinatorial isotopy”, Publ. IHES, 19 (1964), 69–74

[43] Kuperberg K., “2-wild trajectories”, Discrete Contin. Dyn. Syst., 2005, Suppl. vol ., 518–523

[44] Max N. L., “Homeomorphisms of $S^n\times S^1$”, Bull. Am. Math. Soc., 74:6, 939–942

[45] Medvedev T., Pochinka O., “The wild Fox–Artin arc in invariant sets of dynamical systems”, Dyn. Syst., 33:4 (2018), 660–666

[46] Miller R. T., “Approximating codimension 3 embeddings”, Ann. Math. (2), 95:3 (1972), 406–416

[47] Palis J., “On Morse–Smale dynamical systems”, Topology, 8:4 (1969), 385–404

[48] Palis J., “Vector fields generate few diffeomorphisms”, Bull. Am. Math. Soc., 80 (1974), 503–505

[49] Palis J., Smale S., “Structural stability theorem”, Global Analysis, Proc. Symp. Pure Math., 14, American Math. Soc., Providence, 1970

[50] Pixton D., “Wild unstable manifolds”, Topology, 16:2 (1977), 167–172

[51] Pochinka O., “Diffeomorphisms with mildly wild frame of separatrices”, Zesz. Nauk. Uniw. Jagiell, 47 (2009), 149–154

[52] Smale S., “Differentiable dynamical systems”, Bull. Am. Math. Soc., 73:6 (1967), 747–817

[53] Weller G. P., “Locally flat imbeddings of topological manifolds in codimension three”, Trans. Am. Math. Soc., 157 (1971), 161–178

[54] Young G. S., “On the factors and fiberings of manifolds”, Proc. Am. Math. Soc., 1 (1950), 215–223

[55] Zhuzhoma E. V., Medvedev V. S., “Morse–Smale systems with few non-wandering points”, Topology Appl., 160:3 (2013), 498–507