Hostname: page-component-f554764f5-fr72s Total loading time: 0 Render date: 2025-04-17T18:45:50.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Almost existence from the feral perspective and some questions

Part of: Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Symplectic geometry, contact geometry Complex manifolds Smooth dynamical systems: general theory

Published online by Cambridge University Press:  10 June 2021

JOEL W. FISH  and
HELMUT H. W. HOFER Open the ORCID record for HELMUT H. W. HOFER [Opens in a new window]
JOEL W. FISH*
Affiliation:
Department of Mathematics, University of Massachusetts Boston, Boston, MA, USA
HELMUT H. W. HOFER
Affiliation:
School of Mathematics, Institute for Advanced Study, Princeton, NJ, USA (e-mail: [email protected])
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We use feral pseudoholomorphic curves and adiabatic degeneration to prove an extended version of the so-called ‘almost existence result’ for regular compact Hamiltonian energy surfaces. That is, that for a variety of symplectic manifolds equipped with a Hamiltonian, almost every (non-empty) compact energy level has a periodic orbit.

Keywords

feral curvesalmost existenceadiabatic

MSC classification

Primary: 32Q65: Pseudoholomorphic curves 37C99: None of the above, but in this section 37J99: None of the above, but in this section
Secondary: 53D99: None of the above, but in this section
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 2: Anatole Katok Memorial Issue Part 1: Special Issue of Ergodic Theory and Dynamical Systems , February 2022 , pp. 792 - 834
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abbas, C. and Hofer, H.. Holomorphic Curves and Global Questions in Contact Geometry (Birkhäuser Advanced Texts). Birkhäuser, Basel, 2019.10.1007/978-3-030-11803-7CrossRefGoogle Scholar
Asaoka, M. and Irie, K.. A C∞ closing lemma for Hamiltonian diffeomorphisms of closed surfaces. Geom. Funct. Anal. 26(5) (2016), 1245–1254.10.1007/s00039-016-0386-3CrossRefGoogle Scholar
Bartle, R. G.. The Elements of Integration. Wiley, New York, 1966.Google Scholar
Benci, V., Hofer, H. and Rabinowitz, P.. A remark on a priori bounds and existence for periodic solutions of Hamiltonian systems. Periodic Solutions of Hamiltonian Systems and Related Topics (Il Ciocco, 1986) (NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 209). Ed. Rabinowitz, P. H., Ambrosetti, A., Ekeland, I. and Zehnder, E. J.. D. Reidel, Dordrecht, 1987, pp. 85–88.Google Scholar
Cristofaro-Gardiner, D., Hutchings, M. and Ramos, V.. The asymptotics of ECH capacities. Invent. Math. 199(1) (2015), 187–214.10.1007/s00222-014-0510-7CrossRefGoogle Scholar
Eliashberg, Y., Givental, A. and Hofer, H.. Introduction to symplectic field theory. Geom. Funct. Anal. Special Volume, Part II (2000), 560–673. GAFA 2000 (Tel Aviv, 1999).10.1007/978-3-0346-0425-3_4CrossRefGoogle Scholar
Fish, J. W.. Target-local Gromov compactness. Geom. Topol. 15(2) (2011), 765–826.10.2140/gt.2011.15.765CrossRefGoogle Scholar
Fish, J. W. and Hofer, H.. Feral curves and minimal sets. Prepint, 2018, arXiv:1812.06554.Google Scholar
Ginzburg, V. L.. An embedding ${S}^{2n-1}\to {\mathbb{R}}^{2n}$ , $2n-1\ge 7$ , whose Hamiltonian flow has no periodic trajectories. Int. Math. Res. Not. 2 (1995), 83–97.10.1155/S1073792895000079CrossRefGoogle Scholar
Ginzburg, V. L. and Gürel, B. Z.. A C 2-smooth counterexample to the Hamiltonian Seifert conjecture in ℝ4 . Ann. of Math. (2) 158(3) (2003), 953–976.10.4007/annals.2003.158.953CrossRefGoogle Scholar
Herman, M. R.. Exemples de flots hamiltoniens dont aucune perturbation en topologie ${C}^{\infty }$ n’a d’orbites périodiques sur un ouvert de surfaces d’énergies. C. R. Acad. Sci. Paris Sér. I Math. 312(13) (1991), 989–994.Google Scholar
Herman, M. R.. Examples of compact hypersurfaces in ${R}^{2p},2p\ge 6$ , with no periodic orbits. Hamiltonian Systems with Three or More Degrees of Freedom (S’Agaró, 1995) (NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 533). Ed. Simo, C.. Kluwer Academic Publishers, Dordrecht, 1999, p. 126.10.1007/978-94-011-4673-9_11CrossRefGoogle Scholar
Hofer, H.. Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three. Invent. Math. 114(3) (1993), 515–563.10.1007/BF01232679CrossRefGoogle Scholar
Hofer, H. and Viterbo, C.. The Weinstein conjecture in the presence of holomorphic spheres. Comm. Pure Appl. Math. 45(5) (1992), 583–622.10.1002/cpa.3160450504CrossRefGoogle Scholar
Hofer, H. and Zehnder, E.. Periodic solutions on hypersurfaces and a result by C. Viterbo. Invent. Math. 90(1) (1987), 1–9.10.1007/BF01389030CrossRefGoogle Scholar
Hofer, H. and Zehnder, E.. A new capacity for symplectic manifolds. Analysis, Et Cetera. Ed. Rabinowitz, P. and Zehnder, E.. Academic Press, Burlington, MA, 1990, pp. 405–427.10.1016/B978-0-12-574249-8.50023-7CrossRefGoogle Scholar
Hofer, H. and Zehnder, E.. Symplectic Invariants and Hamiltonian Dynamics (Modern Birkhäuser Classics). Birkhäuser Verlag, Basel, 2011. Reprint of the 1994 edition.10.1007/978-3-0348-0104-1CrossRefGoogle Scholar
Irie, K.. Dense existence of periodic Reeb orbits and ECH spectral invariants. J. Mod. Dyn. 9 (2015), 357–363.10.3934/jmd.2015.9.357CrossRefGoogle Scholar
Liu, G. and Tian, G.. Weinstein conjecture and GW-invariants. Commun. Contemp. Math. 2(4) (2000), 405–459.10.1142/S0219199700000256CrossRefGoogle Scholar
Lu, G.. Gromov–Witten invariants and pseudo symplectic capacities. Israel J. Math. 156 (2006), 1–63.10.1007/BF02773823CrossRefGoogle Scholar
Rabinowitz, P.. Periodic solutions of Hamiltonian systems. Comm. Pure Appl. Math. 31(2) (1978), 157–184.10.1002/cpa.3160310203CrossRefGoogle Scholar
Rabinowitz, P.. Periodic solutions of a Hamiltonian system on a prescribed energy surface. J. Differential Equations 33(3) (1979), 336–352.10.1016/0022-0396(79)90069-XCrossRefGoogle Scholar
Rabinowitz, P.. On a theorem of Hofer and Zehnder. Periodic Solutions of Hamiltonian Systems and Related Topics (Il Ciocco, 1986) (NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 209). Ed. Rabinowitz, P. H., Ambrosetti, A., Ekeland, I. and Zehnder, E. J.. D. Reidel, Dordrecht, 1987, pp. 245–254.10.1007/978-94-009-3933-2_22CrossRefGoogle Scholar
Struwe, M.. Existence of periodic solutions of Hamiltonian systems on almost every energy surface. Bol. Soc. Brasil. Mat. (N.S.) 20(2) (1990), 49–58.10.1007/BF02585433CrossRefGoogle Scholar
Taubes, C. H.. The Seiberg–Witten equations and the Weinstein conjecture. Geom. Topol. 11 (2007), 2117–2202.10.2140/gt.2007.11.2117CrossRefGoogle Scholar
Viterbo, C.. A proof of Weinstein’s conjecture in $\mathbb{R}$ 2n . Ann. Inst. H. Poincaré Anal. Non Linéaire 4(4) (1987), 337–356.10.1016/s0294-1449(16)30363-8CrossRefGoogle Scholar
Weinstein, A.. On the hypotheses of Rabinowitz’ periodic orbit theorems. J. Differential Equations 33(3) (1979), 353–358.10.1016/0022-0396(79)90070-6CrossRefGoogle Scholar