A \(C^0\) counterexample to the Arnold conjecture
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The Arnold conjecture states that a Hamiltonian diffeomorphism of a closed and connected symplectic manifold \((M, \omega )\) must have at least as many fixed points as the minimal number of critical points of a smooth function on M. It is well known that the Arnold conjecture holds for Hamiltonian homeomorphisms of closed symplectic surfaces. The goal of this paper is to provide a counterexample to the Arnold conjecture for Hamiltonian homeomorphisms in dimensions four and higher. More precisely, we prove that every closed and connected symplectic manifold of dimension at least four admits a Hamiltonian homeomorphism with a single fixed point.
Keywords\(C^0\) Symplectic geometry Symplectic and Hamiltonian homeomorphisms Arnold conjecture Hamiltonian dynamics
We would like to thank Yasha Eliashberg, Viktor Ginzburg, Helmut Hofer, Rémi Leclercq, Frédéric Le Roux, Patrice Le Calvez, Emmanuel Opshtein, Leonid Polterovich, and Claude Viterbo for fruitful discussions. LB: The research leading to this project began while I was a Professeur Invité at the Université Pierre et Marie Curie. I would like to express my deep gratitude to the members of the Institut Mathématique de Jussieu–Paris Rive Gauche, especially the team Analyse Algébrique, for their warm hospitality. SS: I would like to thank the School of Mathematics at the Institute for Advanced Study and the Department of Mathematics at MIT, where different parts of this project were carried out, for their hospitality. LB was partially supported by the Israel Science Foundation Grant 1380/13, by the Alon Fellowship, and by the Raymond and Beverly Sackler Career Development Chair. VH was partially supported by the Agence Nationale de la Recherche, Projects ANR-11-JS01-010-01 and ANR-12-BS020-0020. SS was partially supported by the NSF Postdoctoral Fellowship Grant No. DMS-1401569.
- 1.Barannikov, S.A.: The framed Morse complex and its invariants. In: Arnold, V.I. (ed.) Singularities and Bifurcations. Transl. ed. by Sossinsky, A.B., vol. 21 of Advances in Soviet Mathematics, pp. 93–115. American Mathematical Society, Providence, RI (1994)Google Scholar
- 7.Eliashberg, Y.: Estimates on the Number of Fixed Points of Area Preserving Transformations. Syktyvkar University, Syktyvkar (1979). (preprint)Google Scholar
- 8.Entov, M., Polterovich, L.: \(C^0\)-rigidity of Poisson brackets. In: Symplectic Topology and Measure Preserving Dynamical Systems, vol. 512 of Contemporary Mathematics, pp. 25–32. American Mathematical Society, Providence, RI (2010)Google Scholar
- 18.Gromov, M.: Soft and hard symplectic geometry. In: Proceedings of the International Congress of Mathematicians, vols. 1, 2 (Berkeley, CA, 1986), pp. 81–98. American Mathematical Society, Providence, RI (1987)Google Scholar
- 21.Howard, W.: Action selectors and the fixed point set of a Hamiltonian diffeomorphism. ArXiv:1211.0580
- 28.Matsumoto, S.: Arnold conjecture for surface homeomorphisms. In: Proceedings of the French–Japanese Conference “Hyperspace Topologies and Applications” (La Bussière, 1997), vol. 104, pp. 191–214 (2000)Google Scholar
- 29.Oh, Y.-G.: Construction of Spectral Invariants of Hamiltonian Paths on Closed Symplectic Manifolds. The Breadth of Symplectic and Poisson Geometry. Progress in Mathematics, vol. 232, pp. 525–570. Birkhauser, Boston (2005)Google Scholar
- 36.Usher, M., Zhang, J.: Persistent homology and Floer–Novikov theory. ArXiv:1502.07928 (2015)
- 38.Viterbo, C.: On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows. Int. Math. Res. Not. 11, 1–9 (2006)Google Scholar