Previous |  Up |  Next

Article

Keywords:
Arnold conjecture; fixed points; Hamiltonian symplectomorphisms
Summary:
In this note we discuss the collection of statements known as Arnold conjecture for Hamiltonian diffeomorphisms of closed symplectic manifolds. We provide an overview of the homological, stable and strong versions of Arnold conjecture for non-degenerate Hamiltonian systems, a few versions of Arnold conjecture for possibly degenerate Hamiltonian systems, the degenerate version of Arnold conjecture for Hamiltonian homeomorphisms and Sandon’s version of Arnold conjecture for contactomorphisms.
References:
[1] Abbondandolo, A.: Morse theory for Hamiltonian systems. Chapman $\&$ Hall/CRC, Res. Notes Math., Boca Raton FL, 2001. MR 1824111
[2] Arnold, V.I.: Mathematical Methods of Classical Mechanics. Grad. Texts in Math., vol. 60, Springer-Verlag, New York, 1989. MR 0997295
[3] Barraud, J.-F.: A Floer fundamental group. Ann. Sci. École Norm. Sup. (4) 51 (3) (2018), 773–809. DOI 10.24033/asens.2366 | MR 3831037
[4] Buhovsky, L., Humilière, V., Seyfaddini, S.: An Arnold-type principle for non-smooth objects. preprint 2019, available at arXiv:1909.07081.
[5] Buhovsky, L., Humilière, V., Seyfaddini, S.: A $C^0$ counterexample to the Arnold conjecture. Invent. Math. 213 (2018), 759–809. DOI 10.1007/s00222-018-0797-x | MR 3827210
[6] Conley, C., Zehnder, E.: The Birkhoff–Lewis fixed point theorem and a conjecture of V.I. Arnold. Invent. Math. 73 (1983), 33–49. DOI 10.1007/BF01393824 | MR 0707347
[7] Damian, M.: On the stable Morse number of a closed manifold. Bull. London Math. Soc. 34 (2002), 420–430. DOI 10.1112/S0024609301008955 | MR 1897421
[8] Dimitroglou Rizell, G., Golovko, R.: The number of Hamiltonian fixed points on symplectically aspherical manifolds. Proceedings of the Geometry-Topology Conference 2016, Gökova Geometry-Topology Conference (GGT), Gökova, 2017, pp. 138–150. MR 3676086
[9] Eliashberg, Y.: Estimates on the number of fixed points of area preserving transformations. Syktyvkar University, preprint 1979.
[10] Filippenko, B., Wehrheim, K.: A polyfold proof of the Arnold conjecture. preprint 2018, available at arXiv:1810.06180. MR 3576532
[11] Fish, J., Hofer, H.: Lectures on Polyfolds and Symplectic Field Theory. preprint 2018, available at arXiv:1808.07147. MR 3676594
[12] Floer, A.: A relative Morse index for the symplectic action. Comm. Pure Appl. Math. 41 (1988), 393–407. DOI 10.1002/cpa.3160410402 | MR 0933228
[13] Floer, A.: Morse theory for Lagrangian intersections. J. Differential Geom. 28 (1988), 513–547. DOI 10.4310/jdg/1214442477 | MR 0965228
[14] Floer, A.: The unregularized gradient flow of the symplectic action. Comm. Pure Appl. Math. 41 (1988), 775–813. DOI 10.1002/cpa.3160410603 | MR 0948771
[15] Floer, A.: Symplectic fixed points and holomorphic spheres. Comm. Math. Phys. 120 (1989), 575–611. DOI 10.1007/BF01260388 | MR 0987770
[16] Floer, A.: Witten’s complex and infinite dimensional Morse theory. Differential Geom. 30 (1989), 207–221. DOI 10.4310/jdg/1214443291 | MR 1001276
[17] Franks, J.: Rotation vectors and fixed points of area preserving surface diffeomorphisms. Trans. Amer. Math. Soc. 348 (7) (1996), 2637–2662. DOI 10.1090/S0002-9947-96-01502-4 | MR 1325916
[18] Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian intersection Floer theory: anomaly and obstruction. Part I. AMS/IP Studies in Advanced Mathematics, American Mathematical Society, Providence, RI; Internationl Press, Somerville, MA, 2009. MR 2553465
[19] Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian intersection Floer theory: anomaly and obstruction. Part II. AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI; Internationl Press, Somerville, MA, 2009. MR 2553465
[20] Fukaya, K., Ono, K.: Arnold conjecture and Gromov-Witten invariant. Topology 38 (5) (1999), 933–1048. DOI 10.1016/S0040-9383(98)00042-1 | MR 1688434
[21] Fukaya, K., Ono, K.: Arnold conjecture and Gromov-Witten invariant for general symplectic manifolds. The Arnoldfest. Proceedings of a conference in honour of V. I. Arnold for his 60th birthday, Toronto, Canada, June 15-21, 1997 (al., E. Bierstone (ed.) et, ed.), vol. 24, 1999. MR 1733575
[22] Gompf, R.E.: A new construction of symplectic manifolds. Ann. of Math. 2 (1995), 527–595. DOI 10.2307/2118554 | MR 1356781
[23] Granja, G., Karshon, Y., Pabiniak, M., Sandon, S.: Givental’s non-linear Maslov index on lens spaces. preprint 2017, available at arXiv:1704.05827.
[24] Gromov, M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82 (2) (1985), 307–347. DOI 10.1007/BF01388806 | MR 0809718
[25] Hofer, H., Wysocki, K., Zehnder, E.: Polyfold and Fredholm Theory. preprint 2017, available at arXiv:1707.08941v1.
[26] Hofer, H., Zehnder, E.: Symplectic invariants and Hamiltonian dynamics. Birkhäuser Verlag, Basel, xiv+341. MR 2797558
[27] Le Calvez, P.: Periodic orbits of Hamiltonian homeomorphisms of surfaces. Duke Math. J. 133 (1) (2006), 125–184. DOI 10.1215/S0012-7094-06-13315-X | MR 2219272
[28] Liu, G., Tian, G.: Floer homology and Arnold conjecture. J. Differential Geom. 49 (1) (1998), 1–74. DOI 10.4310/jdg/1214460936 | MR 1642105
[29] Matsumoto, S.: Arnold conjecture for surface homeomorphisms. Proceedings of the French-Japanese Conference Hyperspace Topologies and Applications, (La Bussiere, 1997), vol. 104, 2000, pp. 191–214. MR 1780985
[30] Ono, K., Pajitnov, A.: On the fixed points of a Hamiltonian diffeomorphism in presence of fundamental group. Essays in Mathematics and its Applications: In Honor of Vladimir Arnold, Springer, 2016, pp. 199–229. MR 3526921
[31] Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. preprint 2002, available at arXiv:0211159. MR 4105148 | Zbl 1130.53001
[32] Piunikhin, S., Salamon, D., Schwarz, M.: Symplectic Floer-Donaldson theory and quantum cohomology. Contact and symplectic geometry (Cambridge, 1994), Publ. Newton Inst., Cambridge Univ. Press, 1996. MR 1432464
[33] Ruan, Y.: Virtual neighborhoods and pseudo-holomorphic curves. Proceedings of 6th Gokova Geometry-Topology Conference, vol. 23, Gokova, 1999. MR 1701645
[34] Rudyak, Yu.B., Oprea, J.: On the Lusternik-Schnirelmann category of symplectic manifolds and the Arnold conjecture. Math. Z. 230 (4) (1999), 673–678. DOI 10.1007/PL00004709 | MR 1686579
[35] Salamon, D.: Lectures on Floer homology. Symplectic Topology, I.A.S./Park City Math. Series. Am. Math. Soc., Providence.
[36] Salamon, D.A., Zehnder, E.: Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. Comm. Pure Appl. Math. 45 (1992), 1303–1360. DOI 10.1002/cpa.3160451004 | MR 1181727
[37] Sandon, S.: On iterated translated points for contactomorphisms of $\mathbb{R}^{2n+1}$ and $\mathbb{R}^{2n} \times S^1$. Internat. J. Math. 23 (2) (2012), 14 pp. MR 2890476
[38] Sandon, S.: A Morse estimate for translated points of contactomorphisms of spheres and projective spaces. Geom. Dedicata 165 (2013), 95–110. DOI 10.1007/s10711-012-9741-1 | MR 3079344
[39] Schwarz, M.: Morse homology. Progress in Math., vol. 111, Birkhäuser, 1993. MR 1239174
[40] Smale, S.: On the structure of manifolds. Amer. J. Math. 84 (1962), 387–399. DOI 10.2307/2372978 | MR 0153022
[41] Tervil, B.: Translated points for prequantization spaces over monotone toric manifolds. preprint 2018, available at arXiv:1811.09984.
Partner of
EuDML logo