On the construction of a $C^2$-counterexample to the Hamiltonian Seifert Conjecture in $\mathbb {R}^4$
Authors:
Viktor L. Ginzburg and Başak Z. Gürel
Journal:
Electron. Res. Announc. Amer. Math. Soc. 8 (2002), 11-19
MSC (2000):
Primary 37J45; Secondary 53D30
DOI:
https://doi.org/10.1090/S1079-6762-02-00100-2
Published electronically:
June 19, 2002
MathSciNet review:
1911741
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Abstract: We outline the construction of a proper $C^2$-smooth function on $\mathbb {R}^4$ such that its Hamiltonian flow has no periodic orbits on at least one regular level set. This result can be viewed as a $C^2$-smooth counterexample to the Hamiltonian Seifert conjecture in dimension four.
- Viktor L. Ginzburg, An embedding $S^{2n-1}\to \textbf {R}^{2n}$, $2n-1\geq 7$, whose Hamiltonian flow has no periodic trajectories, Internat. Math. Res. Notices 2 (1995), 83–97. MR 1317645, DOI 10.1155/S1073792895000079
- Viktor L. Ginzburg, A smooth counterexample to the Hamiltonian Seifert conjecture in $\mathbf R^6$, Internat. Math. Res. Notices 13 (1997), 641–650. MR 1459629, DOI 10.1155/S1073792897000421
- Viktor L. Ginzburg, Hamiltonian dynamical systems without periodic orbits, Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, vol. 196, Amer. Math. Soc., Providence, RI, 1999, pp. 35–48. MR 1736212, DOI 10.1090/trans2/196/03
[Gi4]gi:barcelona V. L. Ginzburg, The Hamiltonian Seifert conjecture: examples and open problems, math.DG/0004020; to appear in Proceedings of the Third ECM, Barcelona, 2000, Birkhäuser.
[GG]GG V. L. Ginzburg and B. Z. Gürel, A $C^2$-smooth counterexample to the Hamiltonian Seifert conjecture in $\mathbb {R}^4$, in preparation.
[He1]herman-fax M.-R. Herman, Fax to Eliashberg, 1994.
- Carles Simó (ed.), Hamiltonian systems with three or more degrees of freedom, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 533, Kluwer Academic Publishers Group, Dordrecht, 1999. MR 1720877, DOI 10.1007/978-94-011-4673-9
- H. Hofer and E. Zehnder, Periodic solutions on hypersurfaces and a result by C. Viterbo, Invent. Math. 90 (1987), no. 1, 1–9. MR 906578, DOI 10.1007/BF01389030
- Helmut Hofer and Eduard Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1994. MR 1306732, DOI 10.1007/978-3-0348-8540-9
- Jun Hu and Dennis P. Sullivan, Topological conjugacy of circle diffeomorphisms, Ergodic Theory Dynam. Systems 17 (1997), no. 1, 173–186. MR 1440774, DOI 10.1017/S0143385797061002
- Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
[Ke]ely:example E. Kerman, New smooth counterexamples to the Hamiltonian Seifert conjecture, Preprint 2001, math.DG/0101185; to appear in the Journal of Symplectic Geometry.
- Greg Kuperberg, A volume-preserving counterexample to the Seifert conjecture, Comment. Math. Helv. 71 (1996), no. 1, 70–97. MR 1371679, DOI 10.1007/BF02566410
- Greg Kuperberg and Krystyna Kuperberg, Generalized counterexamples to the Seifert conjecture, Ann. of Math. (2) 143 (1996), no. 3, 547–576. MR 1394969, DOI 10.2307/2118536
- Krystyna Kuperberg, A smooth counterexample to the Seifert conjecture, Ann. of Math. (2) 140 (1994), no. 3, 723–732. MR 1307902, DOI 10.2307/2118623
- Krystyna Kuperberg, Counterexamples to the Seifert conjecture, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 831–840. MR 1648130
- Krystyna Kuperberg, Aperiodic dynamical systems, Notices Amer. Math. Soc. 46 (1999), no. 9, 1035–1040. MR 1710669
- Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. Oxford Science Publications. MR 1373431
- Paul A. Schweitzer, Counterexamples to the Seifert conjecture and opening closed leaves of foliations, Ann. of Math. (2) 100 (1974), 386–400. MR 356086, DOI 10.2307/1971077
- Michael Struwe, Existence of periodic solutions of Hamiltonian systems on almost every energy surface, Bol. Soc. Brasil. Mat. (N.S.) 20 (1990), no. 2, 49–58. MR 1143173, DOI 10.1007/BF02585433
- F. Wesley Wilson Jr., On the minimal sets of non-singular vector fields, Ann. of Math. (2) 84 (1966), 529–536. MR 202155, DOI 10.2307/1970458
[Gi1]gi:seifert V. L. Ginzburg, An embedding $S^{2n-1}\to \mathbb {R}^{2n}$, $2n-1\geq 7$, whose Hamiltonian flow has no periodic trajectories, IMRN 1995, no. 2, 83–98.
[Gi2]gi:seifert97 V. L. Ginzburg, A smooth counterexample to the Hamiltonian Seifert conjecture in $\mathbb {R}^6$, IMRN 1997, no. 13, 641–650.
[Gi3]gi:bayarea V. L. Ginzburg, Hamiltonian dynamical systems without periodic orbits, in Northern California Symplectic Geometry Seminar, pp. 35–48, Amer. Math. Soc. Transl. Ser. 2, vol. 196, Amer. Math. Soc., Providence, RI, 1999.
[Gi4]gi:barcelona V. L. Ginzburg, The Hamiltonian Seifert conjecture: examples and open problems, math.DG/0004020; to appear in Proceedings of the Third ECM, Barcelona, 2000, Birkhäuser.
[GG]GG V. L. Ginzburg and B. Z. Gürel, A $C^2$-smooth counterexample to the Hamiltonian Seifert conjecture in $\mathbb {R}^4$, in preparation.
[He1]herman-fax M.-R. Herman, Fax to Eliashberg, 1994.
[He2]herman M.-R. Herman, Examples of compact hypersurfaces in $\mathbb {R}^{2p}$, $2p\geq 6$, with no periodic orbits, in Hamiltonian systems with three or more degrees of freedom, C. Simo (Editor), NATO Adv. Sci. Inst. Ser. C, Math. Phys. Sci., vol. 533, Kluwer Acad. Publ., Dordrecht, 1999.
[HZ1]ho-ze:per-sol H. Hofer and E. Zehnder, Periodic solution on hypersurfaces and a result by C. Viterbo, Invent. Math. 90 (1987), 1–9.
[HZ2]ho-ze:book H. Hofer and E. Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser, Advanced Texts; Basel-Boston-Berlin, 1994.
[HS]HS J. Hu and D. Sullivan, Topological conjugacy of circle diffeomorphisms, Ergodic Theory Dynam. Systems 17 (1997), 173–186.
[KH]KH A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54. Cambridge University Press, Cambridge, 1995.
[Ke]ely:example E. Kerman, New smooth counterexamples to the Hamiltonian Seifert conjecture, Preprint 2001, math.DG/0101185; to appear in the Journal of Symplectic Geometry.
[KuG]kug G. Kuperberg, A volume-preserving counterexample to the Seifert conjecture, Comment. Math. Helv. 71 (1996), 70–97.
[KuGK]kugk G. Kuperberg and K. Kuperberg, Generalized counterexamples to the Seifert conjecture, Ann. Math. 144 (1996), 239–268.
[KuK1]kuk K. Kuperberg, A smooth counterexample to the Seifert conjecture in dimension three, Ann. Math. 140 (1994), 723–732.
[KuK2]kuk:icm K. Kuperberg, Counterexamples to the Seifert conjecture, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. (1998), Extra Vol. II, 831–840.
[KuK3]kuk:notices K. Kuperberg, Aperiodic dynamical systems, Notices Amer. Math. Soc. 46 (1999), 1035–1040.
[McDS]mcduff-sal D. McDuff and D. Salamon, Introduction to symplectic topology, Oxford Mathematical Monographs, Oxford University Press, New York, 1995.
[Sc]schweitzer P.A. Schweitzer, Counterexamples to the Seifert conjecture and opening closed leaves of foliations, Ann. Math. 100 (1970), 229–234.
[St]str M. Struwe, Existence of periodic solutions of Hamiltonian systems on almost every energy surface, Bol. Soc. Bras. Mat. 20 (1990), 49–58.
[Wi]wilson F. Wilson, On the minimal sets of nonsingular vector fields, Ann. Math. 84 (1966), 529–536.
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Additional Information
Viktor L. Ginzburg
Affiliation:
Department of Mathematics, UC Santa Cruz, Santa Cruz, CA 95064, USA
Email:
ginzburg@math.ucsc.edu
Başak Z. Gürel
Affiliation:
Department of Mathematics, UC Santa Cruz, Santa Cruz, CA 95064, USA
Email:
basak@math.ucsc.edu
Keywords:
Hamiltonian Seifert conjecture,
periodic orbits
Received by editor(s):
September 20, 2001
Published electronically:
June 19, 2002
Additional Notes:
The work is partially supported by the NSF and by the faculty research funds of the University of California, Santa Cruz.
Communicated by:
Krystyna Kuperberg
Article copyright:
© Copyright 2002
American Mathematical Society
