×

Morse-Smale diffeomorphisms with non-wandering points of pairwise different Morse indices on 3-manifolds. (English. Russian original) Zbl 1548.37031

Russ. Math. Surv. 79, No. 1, 135-184 (2024); translation from Usp. Mat. Nauk 79, No. 1, 135-184 (2024).
Summary: 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 S. Smale [Bol. Soc. Mat. Mex., II. Ser. 5, 195–198 (1960; Zbl 0121.17906)] 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 Ch. Bonatti et al. [Ergodic Theory Dyn. Syst. 39, No. 9, 2403–2432 (2019; Zbl 1431.37032)], such diffeomorphisms have no energy function and the question of the topology of the supporting manifold is still open. According to results due to V. Z. Grines et al. [Sb. Math. 194, No. 7, 979–1007 (2003; Zbl 1077.37025); translation from Mat. Sb. 194, No. 7, 25–56 (2003); Proc. Steklov Inst. Math. 271, 103–124 (2010; Zbl 1302.37021); translation from Tr. Mat. Inst. Steklova 271, 111–133 (2010); J. Math. Sci., New York 253, No. 5, 676–691 (2021; Zbl 1471.76090); translation from Sovrem. Mat., Fundam. Napravl. 63, No. 3, 455–474 (2017)], \(M^3\) 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 C. Bonatti and V. Grines [J. Dyn. Control Syst. 6, No. 4, 579–602 (2000; Zbl 0959.37017)].

MSC:

37C05 Dynamical systems involving smooth mappings and diffeomorphisms
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37C20 Generic properties, structural stability of dynamical systems
37D15 Morse-Smale systems

References:

[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. · Zbl 1099.37518 · doi:10.1142/S0218127404009806
[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. · Zbl 1478.37034 · doi:10.1007/s12346-021-00518-1
[3] A. Andronov and L. S. Pontryagin, “Coarse systems”, Dokl. Akad. Nauk SSSR 14:5 (1937), 247-250. (Russian) · Zbl 0016.11301
[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. · Zbl 0805.58013
[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. · Zbl 0803.58007
[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. · Zbl 0805.58014
[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. · Zbl 0959.37017 · doi:10.1023/A:1009508728879
[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. · Zbl 1431.37032 · doi:10.1017/etds.2017.129
[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. · Zbl 1011.37013
[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. · Zbl 1157.37309 · doi:10.1016/S0166-8641(01)00028-1
[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. · Zbl 1056.37023 · doi:10.1016/S0040-9383(03)00053-3
[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. · Zbl 1282.37022
[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. · Zbl 1377.37044 · doi:10.1134/S0081543817040034
[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. · Zbl 1435.37050 · doi:10.1215/00127094-2019-0019
[15] C. Bonatti and R. Langevin, Difféomorphismes de Smale des surfaces, With the collaboration of E. Jeandenans, Astérisque, vol. 250, Soc. Math. France, Paris 1998, viii+235 pp. · Zbl 0922.58058
[16] G. Fleitas, “Classification of gradient-like flows on dimensions two and three”, Bol. Soc. Brasil. Mat. 6:2 (1975), 155-183. · Zbl 0383.58013 · doi:10.1007/BF02584782
[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. · Zbl 0814.58025 · doi:10.1007/BF01209552
[18] V. Z. Grines and E. Ya. Gurevich, “A combinatorial invariant of gradient-like flows on a connected sum of S n-1 × S 1 ”, Sb. Math. 214:5 (2023), 703-731. · Zbl 1533.37063 · doi:10.4213/sm9761e
[19] V. Z. Grines and E. Ya. Gurevich, “Topological classification of flows without heteroclinic intersections on a connected sum of manifolds S n-1 × S 1 ”, Russian Math. Surveys 77:4 (2022), 759-761. · Zbl 1526.37025 · doi:10.4213/rm10047e
[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. · Zbl 1370.37027 · doi:10.1134/S0001434614110297
[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. · Zbl 1376.37067 · doi:10.1070/RM9747
[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. · Zbl 1444.37002 · doi:10.1070/RM9855
[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. · Zbl 0829.57015 · doi:10.1070/RM1994v049n02ABEH002220
[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. · Zbl 1351.37108 · doi:10.1070/SM2014v205n10ABEH004423
[25] V. Z. Grines, T. V. Medvedev, and O. V. Pochinka, Dynamical systems on 2-and 3-manifolds, Dev. Math., vol. 46, Springer, Cham 2016, xxvi+295 pp. · Zbl 1417.37018 · doi:10.1007/978-3-319-44847-3
[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. · Zbl 1364.76238 · doi:10.1016/j.physd.2014.11.004
[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., vol. 2, Springer, Heidelberg 2011, pp. 403-427. · Zbl 1254.37025 · doi:10.1007/978-3-642-14788-3_31
[28] V. Z. Grines and O. V. Pochinka, “Morse-Smale cascades on 3-manifolds”, Russian Math. Surveys 68:1 (2013), 117-173. · Zbl 1277.37055 · doi:10.1070/RM2013v068n01ABEH004823
[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. · Zbl 1077.37025 · doi:10.1070/SM2003v194n07ABEH000751
[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. · Zbl 1302.37021 · doi:10.1134/S0081543810040097
[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. · Zbl 1471.76090 · doi:10.1007/s10958-021-05261-1
[32] W. Hurewicz and H. Wallman, Dimension theory, Princeton Univ. Press, Princeton, NJ 1948, vii+165 pp. · Zbl 0036.12501
[33] P. Kirk and C. Livingston, “Knot invariants in 3-manifolds and essential tori”, Pacific J. Math. 197:1 (2001), 73-96. · Zbl 1045.57004 · doi:10.2140/pjm.2001.197.73
[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. · Zbl 1481.37023 · doi:10.1134/S1560354720060143
[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. · Zbl 1013.37024
[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 0016.11302
[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) · Zbl 0064.33903
[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. · Zbl 1465.37028 · doi:10.1063/5.0029352
[39] B. Mazur, “A note on some contractible 4-manifolds”, Ann. of Math. (2) 73:1 (1961), 221-228. · Zbl 0127.13604 · doi:10.2307/1970288
[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. · Zbl 0161.20302
[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. · Zbl 1254.37019 · doi:10.1134/S0081543810030156
[42] J. Munkres, “Obstructions to the smoothing of piecewise-differentiable homeomorphisms”, Ann. of Math. (2) 72:3 (1960), 521-554. · Zbl 0108.18101 · doi:10.2307/1970228
[43] W. D. Neumann, “Notes on geometry and 3-manifolds”, Low dimensional topology (Eger, 1996/Budapest 1998), Bolyai Soc. Math. Stud., vol. 8, János Bolyai Math. Soc., Budapest 1999, pp. 191-267. · Zbl 0944.57012
[44] A. A. Oshemkov and V. V. Sharko, “Classification of Morse-Smale flows on two-dimensional manifolds”, Sb. Math. 189:8 (1998), 1205-1250. · Zbl 0915.58045 · doi:10.1070/sm1998v189n08ABEH000341
[45] J. Palis, “On Morse-Smale dynamical systems”, Topology 8:4 (1969), 385-404. · Zbl 0189.23902 · doi:10.1016/0040-9383(69)90024-X
[46] J. Palis and S. Smale, “Structural stability theorems”, Global analysis (Berkeley, CA 1968), Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, RI 1970, pp. 223-231. · Zbl 0214.50702
[47] M. M. Peixoto, “Structural stability on two-dimensional manifolds”, Topology 1:2 (1962), 101-120. · Zbl 0107.07103 · doi:10.1016/0040-9383(65)90018-2
[48] M. Peixoto, “Structural stability on two-dimensional manifolds: a further remark”, Topology 2:1-2 (1963), 179-180. · Zbl 0116.06802 · doi:10.1016/0040-9383(63)90032-6
[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, pp. 389-419. · Zbl 0299.58011 · doi:10.1016/B978-0-12-550350-1.50033-3
[50] S. Ju. Piljugin, “Phase diagrams determining Morse-Smale systems without periodic trajectories on spheres”, Differ. Equ. 14:2 (1978), 170-177. · Zbl 0409.34041
[51] D. Pixton, “Wild unstable manifolds”, Topology 16:2 (1977), 167-172. · Zbl 0355.58004 · doi:10.1016/0040-9383(77)90014-3
[52] O. Pochinka, “Diffeomorphisms with mildly wild frame of separatrices”, Univ. Iagel. Acta Math. 47 (2009), 149-154. · Zbl 1223.57029
[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. · Zbl 1511.37032 · doi:10.1134/S0001434622090127
[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. · Zbl 1498.37029 · doi:10.1088/1361-6544/ac4c2c
[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. · Zbl 1516.37036 · doi:10.1134/S0040577923050112
[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. · Zbl 1533.37064 · doi:10.4213/sm9814e
[58] E. R. Priest, Solar magneto-hydrodynamics, D. Reidel Publishing Co., Dordrecht 1982, xix+469 pp. · doi:10.1007/978-94-009-7958-1
[59] E. Priest and T. Forbes, Magnetic reconnection. MHD theory and applications, Cambridge Univ. Press, Cambridge 2000, xii+600 pp. · Zbl 0959.76002 · doi:10.1017/CBO9780511525087
[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. · Zbl 1152.57029 · doi:10.1007/s10958-007-0287-y
[61] D. Rolfsen, Knots and links, Corr. reprint of the 1976 original, Math. Lecture Ser., vol. 7, Publish or Perish, Inc., Houston, TX 1990, xiv+439 pp. · Zbl 0854.57002
[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:10.18500/0869-6632-2021-29-6-863-868
[64] S. Smale, “Morse inequalities for a dynamical system”, Bull. Amer. Math. Soc. 66 (1960), 43-49. · Zbl 0100.29701 · doi:10.1090/S0002-9904-1960-10386-2
[65] Ya. L. Umanskiȋ, “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. · Zbl 0717.58033 · doi:10.1070/SM1991v069n01ABEH001235
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.