Morse-Smale diffeomorphisms with non-wandering points of pairwise different Morse indices on 3-manifolds

O. V. Pochinka   ; E. A. Talanova

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 79 (2024) no. 1, pp. 127-171

Résumé

We consider the class $G$ of orientation-preserving Morse–Smale diffeomorphisms $f$ defined on a closed 3-manifold $M^3$, whose non-wandering set consists of four fixed points with pairwise different Morse indices. It follows from results due to Smale and Meyer that all gradient-like flows with similar properties have a Morse energy function with four critical points of pairwise distinct Morse indices. This means that the supporting manifold $M^3$ of such a flow admits a Heegaard decomposition of genus 1, so that it is diffeomorphic to a lens space $L_{p,q}$. Despite the simple structure of the non-wandering set of diffeomorphisms in the class $G$, there are diffeomorphisms with wildly embedded separatrices. According to results due to Grines, Laudenbach, and Pochinka, such diffeomorphisms have no energy function and the question of the topology of the supporting manifold is still open. According to results due to Grines, Zhuzhoma, and V. Medvedev, $M^3$ is homeomorphic to a lens space $L_{p,q}$ in the case of a tame embedding of the one-dimensional separatrices of the diffeomorphism $f\in G$. Moreover, the wandering set of $f$ contains at least $p$ non-compact heteroclinic curves. We obtain a similar result for arbitrary diffeomorphisms in the class $G$. On each lens space $L_{p,q}$ we construct diffeomorphisms from $G$ with wild embeddings of one-dimensional separatrices. Such examples were previously known only on the 3-sphere. We also show that the topological conjugacy of two diffeomorphisms in $G$ with a unique non-compact heteroclinic curve is fully determined by the equivalence of the Hopf knots that are the projections of one-dimensional saddle separatrices onto the orbit space of the sink basin. Moreover, each Hopf knot $L$ can be realized by such a diffeomorphism. In this sense the result obtained is similar to the classification of Pixton diffeomorphisms obtained by Bonatti and Grines. Bibliography: 65 titles.

Détail

Keywords: Morse–Smale diffeomorphism, heteroclinic curve, Hopf knot, wild embedding.

Bibliographie

[1] V. S. Afraimovich, M. I. Rabinovich, and P. Varona, “Heteroclinic contours in neural ensembles and the winnerless competition principle”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14:4 (2004), 1195–1208 | DOI | MR | Zbl

[2] P. M. Akhmet'ev, T. V. Medvedev, and O. V. Pochinka, “On the number of the classes of topological conjugacy of Pixton diffeomorphisms”, Qual. Theory Dyn. Syst., 20:3 (2021), 76, 15 pp. | DOI | MR | Zbl

[3] A. Andronov and L. S. Pontryagin, “Coarse systems”, Dokl. Akad. Nauk SSSR, 14:5 (1937), 247–250 (Russian) | Zbl

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

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

[6] A. N. Bezdenezhnykh and V. Z. Grines, “Dynamical properties and topological classification of gradient-like diffeomorphisms on two-dimensional manifolds. II”, Selecta Math. Soviet., 11:1 (1992), 13–17 | MR | Zbl

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

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

[9] Ch. Bonatti, V. Z. Grines, V. S. Medvedev, and E. Pécou, “On Morse–Smale diffeomorphisms without heteroclinic intersections on three-manifolds”, Proc. Steklov Inst. Math., 236 (2002), 58–69 | MR | Zbl

[10] C. Bonatti, V. Grines, V. Medvedev, and E. Pécou, “Three-manifolds admitting Morse–Smale diffeomorphisms without heteroclinic curves”, Topology Appl., 117:3 (2002), 335–344 | DOI | MR | Zbl

[11] C. Bonatti, V. Grines, V. Medvedev, and E. Pécou, “Topological classification of gradient-like diffeomorphisms on 3-manifolds”, Topology, 43:2 (2004), 369–391 | DOI | MR | Zbl

[12] Ch. Bonatti, V. Z. Grines, and O. V. Pochina, “Classification of Morse–Smale diffeomorphisms with finite sets of heteroclinic orbits on 3-manifolds”, Dokl. Math., 69:3 (2004), 385–387 | MR | Zbl

[13] Ch. Bonatti, V. Z. Grines, and O. V. Pochinka, “Realization of Morse–Smale diffeomorphisms on 3-manifolds”, Proc. Steklov Inst. Math., 297 (2017), 35–49 | DOI | MR | Zbl

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

[15] C. Bonatti and R. Langevin, Difféomorphismes de Smale des surfaces, With the collaboration of E. Jeandenans, Astérisque, 250, Soc. Math. France, Paris, 1998, viii+235 pp. | MR | Zbl

[16] G. Fleitas, “Classification of gradient-like flows on dimensions two and three”, Bol. Soc. Brasil. Mat., 6:2 (1975), 155–183 | DOI | MR | Zbl

[17] V. Z. Grines, “Topological classification of Morse–Smale diffeomorphisms with finite set of heteroclinic trajectories on surfaces”, Math. Notes, 54:3 (1993), 881–889 | DOI | MR | Zbl

[18] V. Z. Grines and E. Ya. Gurevich, “A combinatorial invariant of gradient-like flows on a connected sum of $\mathbb{S}^{n-1}\times\mathbb{S}^1$”, Sb. Math., 214:5 (2023), 703–731 | DOI | MR

[19] V. Z. Grines and E. Ya. Gurevich, “Topological classification of flows without heteroclinic intersections on a connected sum of manifolds $\mathbb{S}^{n-1}\times\mathbb{S}^{1}$”, Russian Math. Surveys, 77:4 (2022), 759–761 | DOI | MR | Zbl

[20] V. Z. Grines, E. Ya. Gurevich, and O. V. Pochinka, “The energy function of gradient-like flows and the topological classification problem”, Math. Notes, 96:6 (2014), 921–927 | DOI | MR | Zbl

[21] V. Z. Grines, E. Ya. Gurevich, and O. V. Pochinka, “On embedding Morse–Smale diffeomorphisms on the sphere in topological flows”, Russian Math. Surveys, 71:6 (2016), 1146–1148 | DOI | MR | Zbl

[22] V. Z. Grines, E. Ya. Gurevich, E. V. Zhuzhoma, and O. V. Pochinka, “Classification of Morse–Smale systems and topological structure of the underlying manifolds”, Russian Math. Surveys, 74:1 (2019), 37–110 | DOI | MR | Zbl

[23] V. Z. Grines and Kh. Kh. Kalai, “On the topological classification of gradient-like diffeomorphisms on irreducible three-dimensional manifolds”, Russian Math. Surveys, 49:2 (1994), 157–158 | DOI | MR | Zbl

[24] V. Z. Grines, S. Kh. Kapkaeva, and O. V. Pochinka, “A three-colour graph as a complete topological invariant for gradient-like diffeomorphisms of surfaces”, Sb. Math., 205:10 (2014), 1387–1412 | DOI | MR | Zbl

[25] V. Z. Grines, T. V. Medvedev, and O. V. Pochinka, Dynamical systems on 2- and 3-manifolds, Dev. Math., 46, Springer, Cham, 2016, xxvi+295 pp. | DOI | MR | Zbl

[26] V. Grines, T. Medvedev, O. Pochinka, and E. Zhuzhoma, “On heteroclinic separators of magnetic fields in electrically conducting fluids”, Phys. D, 294 (2015), 1–5 | DOI | MR | Zbl

[27] V. Grines and O. Pochinka, “On topological classification of Morse–Smale diffeomorphisms”, Dynamics, games and science. II (Univ. of Minho, Braga 2008), Springer Proc. Math., 2, Springer, Heidelberg, 2011, 403–427 | DOI | MR | Zbl

[28] V. Z. Grines and O. V. Pochinka, “Morse–Smale cascades on 3-manifolds”, Russian Math. Surveys, 68:1 (2013), 117–173 | DOI | MR | Zbl

[29] V. Z. Grines, E. V. Zhuzhoma, and V. S. Medvedev, “New relations for Morse–Smale systems with trivially embedded one-dimensional separatrices”, Sb. Math., 194:7 (2003), 979–1007 | DOI | MR | Zbl

[30] V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev, and O. V. Pochinka, “Global attractor and repeller of Morse–Smale diffeomorphisms”, Proc. Steklov Inst. Math., 271 (2010), 103–124 | DOI | MR | Zbl

[31] V. Z. Grines, E. V. Zhuzhoma, and O. V. Pochinka, “Dynamical systems and topology of magnetic fields in conducting medium”, J. Math. Sci. (N. Y.), 253:5 (2021), 676–691 | DOI | MR | Zbl

[32] W. Hurewicz and H. Wallman, Dimension theory, Princeton Univ. Press, Princeton, NJ, 1948, vii+165 pp. | MR | Zbl

[33] P. Kirk and C. Livingston, “Knot invariants in 3-manifolds and essential tori”, Pacific J. Math., 197:1 (2001), 73–96 | DOI | MR | Zbl

[34] V. E. Kruglov, D. S. Malyshev, O. V. Pochinka, and D. D. Shubin, “On topological classification of gradient-like flows on an $n$-sphere in the sense of topological conjugacy”, Regul. Chaotic Dyn., 25:6 (2020), 716–728 | DOI | MR | Zbl

[35] E. V. Kruglov and E. A. Talanova, “On the realization of Morse–Smale diffeomorphisms with heteroclinic curves on a 3-sphere”, Proc. Steklov Inst. Math., 236 (2002), 201–205 | MR | Zbl

[36] E. A. Leontovich and A. G. Mayer, “One trajectories determining the qualitative structure of the partition of a sphere into trajectories”, Dokl. Akad. Nauk SSSR, 14:5 (1937), 251–257 (Russian) | Zbl

[37] E. A. Leontovich and A. G. Mayer, “A scheme determining the topological structure of a partition into trajectories”, Dokl. Akad. Nauk SSSR, 103:4 (1955), 557–560 (Russian) | MR | Zbl

[38] D. Malyshev, A. Morozov, and O. Pochinka, “Combinatorial invariant for Morse–Smale diffeomorphisms on surfaces with orientable heteroclinic”, Chaos, 31:2 (2021), 023119, 17 pp. | DOI | MR | Zbl

[39] B. Mazur, “A note on some contractible 4-manifolds”, Ann. of Math. (2), 73:1 (1961), 221–228 | DOI | MR | Zbl

[40] J. Milnor, Lectures on the $h$-cobordism theorem, Notes by L. Siebenmann and J. Sondow, Princeton Univ. Press, Princeton, NJ, 1965, v+116 pp. | MR | Zbl

[41] T. M. Mitryakova and O. V. Pochinka, “Necessary and sufficient conditions for the topological conjugacy of surface diffeomorphisms with a finite number of orbits of heteroclinic tangency”, Proc. Steklov Inst. Math., 270 (2010), 194–215 | DOI | MR | Zbl

[42] J. Munkres, “Obstructions to the smoothing of piecewise-differentiable homeomorphisms”, Ann. of Math. (2), 72:3 (1960), 521–554 | DOI | MR | Zbl

[43] W. D. Neumann, “Notes on geometry and 3-manifolds”, Low dimensional topology (Eger, 1996/Budapest 1998), Bolyai Soc. Math. Stud., 8, János Bolyai Math. Soc., Budapest, 1999, 191–267 | MR | Zbl

[44] A. A. Oshemkov and V. V. Sharko, “Classification of Morse–Smale flows on two-dimensional manifolds”, Sb. Math., 189:8 (1998), 1205–1250 | DOI | MR | Zbl

[45] J. Palis, “On Morse–Smale dynamical systems”, Topology, 8:4 (1969), 385–404 | DOI | MR | Zbl

[46] J. Palis and S. Smale, “Structural stability theorems”, Global analysis (Berkeley, CA 1968), Proc. Sympos. Pure Math., 14, Amer. Math. Soc., Providence, RI, 1970, 223–231 | MR | Zbl

[47] M. M. Peixoto, “Structural stability on two-dimensional manifolds”, Topology, 1:2 (1962), 101–120 | DOI | MR | Zbl

[48] M. Peixoto, “Structural stability on two-dimensional manifolds: a further remark”, Topology, 2:1-2 (1963), 179–180 | DOI | MR | Zbl

[49] M. M. Peixoto, “On the classification of flows on 2-manifolds”, Dynamical systems (Univ. Bahia, Salvador 1971), Academic Press, Inc., New York–London, 1973, 389–419 | DOI | MR | Zbl

[50] S. Ju. Piljugin, “Phase diagrams determining Morse–Smale systems without periodic trajectories on spheres”, Differ. Equ., 14:2 (1978), 170–177 | MR | Zbl

[51] D. Pixton, “Wild unstable manifolds”, Topology, 16:2 (1977), 167–172 | DOI | MR | Zbl

[52] O. Pochinka, “Diffeomorphisms with mildly wild frame of separatrices”, Univ. Iagel. Acta Math., 47 (2009), 149–154 | MR | Zbl

[53] O. V. Pochinka and D. D. Shubin, “Nonsingular Morse–Smale flows with three periodic orbits on orientable 3-manifolds”, Math. Notes, 112:3 (2022), 436–450 | DOI | MR | Zbl

[54] O. V. Pochinka and D. D. Shubin, “Non-singular Morse–Smale flows on $n$-manifolds with attractor-repeller dynamics”, Nonlinearity, 35:3 (2022), 1485–1499 | DOI | MR | Zbl

[55] O. V. Pochinka and E. A. Talanova, “Minimizing the number of heteroclinic curves of a 3-diffeomorphism with fixed points with pairwise different Morse indices”, Theoret. and Math. Phys., 215:2 (2023), 729–734 | DOI | MR | Zbl

[56] O. Pochinka and E. Talanova, On the topology of 3-manifolds admitting Morse–Smale diffeomorphisms with four fixed points of pairwise different Morse indices, Cornell Univ., Working paper, 2023, 30 pp., arXiv: 2306.02814

[57] O. Pochinka, E. Talanova, and D. Shubin, “Knot as a complete invariant of a Morse–Smale 3-diffeomorphism with four fixed points”, Sb. Math., 214:8 (2023), 1140–1152 | DOI | MR | Zbl

[58] E. R. Priest, Solar magneto-hydrodynamics, D. Reidel Publishing Co., Dordrecht, 1982, xix+469 pp. | DOI

[59] E. Priest and T. Forbes, Magnetic reconnection. MHD theory and applications, Cambridge Univ. Press, Cambridge, 2000, xii+600 pp. | DOI | MR | Zbl

[60] A. Prishlyak, “Complete topological invariants of Morse–Smale flows and handle decompositions of 3-manifolds”, J. Math. Sci. (N.Y.), 144:5 (2007), 4492–4499 | DOI | MR | Zbl

[61] D. Rolfsen, Knots and links, Math. Lecture Ser., 7, Corr. reprint of the 1976 original, Publish or Perish, Inc., Houston, TX, 1990, xiv+439 pp. | MR | Zbl

[62] V. I. Shmukler and O. V. Pochinka, “Bifurcations changing the type of heteroclinic surves of a Morse–Smale 3-diffeomorphism”, Taurida J. Comput. Sci. Theory Math., 2021, no. 1, 101–114 (Russian)

[63] D. D. Shubin, “Topology of supporting manifolds of non-singular flows with three periodic orbits”, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelinein. Din., 29:6 (2021), 863–868 | DOI

[64] S. Smale, “Morse inequalities for a dynamical system”, Bull. Amer. Math. Soc., 66 (1960), 43–49 | DOI | MR | Zbl

[65] Ya. L. Umanskiĭ, “Necessary and sufficient conditions for topological equivalence of three-dimensional Morse–Smale dynamical systems with a finite number of singular trajectories”, Math. USSR-Sb., 69:1 (1991), 227–253 | DOI | MR | Zbl