Volume 9, issue 4 (2005)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 23
Issue 2, 541–1084
Issue 1, 1–540

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Subscriptions
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Ethics Statement
Author Index
To Appear
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Other MSP Journals
Squeezing in Floer theory and refined Hofer–Zehnder capacities of sets near symplectic submanifolds

Ely Kerman

Geometry & Topology 9 (2005) 1775–1834
Bibliography
1 V I Arnol’d, On some problems in symplectic topology, from: "Topology and geometry—Rohlin Seminar", Lecture Notes in Math. 1346, Springer (1988) 1 MR970068
2 V Benci, H Hofer, P H Rabinowitz, A remark on a priori bounds and existence for periodic solutions of Hamiltonian systems, from: "Periodic solutions of Hamiltonian systems and related topics (Il Ciocco, 1986)", NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 209, Reidel (1987) 85 MR920609
3 V Benci, P H Rabinowitz, A priori bounds for periodic solutions of a class of Hamiltonian systems, Ergodic Theory Dynam. Systems 8* (1988) 27 MR967627
4 M Bialy, L Polterovich, Geodesics of Hofer's metric on the group of Hamiltonian diffeomorphisms, Duke Math. J. 76 (1994) 273 MR1301192
5 P Biran, Lagrangian barriers and symplectic embeddings, Geom. Funct. Anal. 11 (2001) 407 MR1844078
6 P Biran, L Polterovich, D Salamon, Propagation in Hamiltonian dynamics and relative symplectic homology, Duke Math. J. 119 (2003) 65 MR1991647
7 S V Bolotin, Libration motions of natural dynamical systems, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1978) 72 MR524544
8 R Bott, L W Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer (1982) MR658304
9 G Contreras, L Macarini, G P Paternain, Periodic orbits for exact magnetic flows on surfaces, Int. Math. Res. Not. (2004) 361 MR2036336
10 K Cieliebak, A Floer, H Hofer, Symplectic homology II: A general construction, Math. Z. 218 (1995) 103 MR1312580
11 K Cieliebak, A Floer, H Hofer, K Wysocki, Applications of symplectic homology II: Stability of the action spectrum, Math. Z. 223 (1996) 27 MR1408861
12 K Cieliebak, V L Ginzburg, E Kerman, Symplectic homology and periodic orbits near symplectic submanifolds, Comment. Math. Helv. 79 (2004) 554 MR2081726
13 C C Conley, E Zehnder, The Birkhoff–Lewis fixed point theorem and a conjecture of V I: Arnol'd, Invent. Math. 73 (1983) 33 MR707347
14 Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. (2000) 560 MR1826267
15 A Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988) 513 MR965228
16 A Floer, Witten's complex and infinite-dimensional Morse theory, J. Differential Geom. 30 (1989) 207 MR1001276
17 A Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989) 575 MR987770
18 A Floer, H Hofer, Symplectic homology I: Open sets in $\mathbb{C}^n$, Math. Z. 215 (1994) 37 MR1254813
19 A Floer, H Hofer, K Wysocki, Applications of symplectic homology I, Math. Z. 217 (1994) 577 MR1306027
20 A Floer, H Hofer, C Viterbo, The Weinstein conjecture in $P\times \mathbb{C}^l$, Math. Z. 203 (1990) 469 MR1038712
21 V L Ginzburg, New generalizations of Poincaré's geometric theorem, Funktsional. Anal. i Prilozhen. 21 (1987) 16, 96 MR902290
22 V L Ginzburg, On closed trajectories of a charge in a magnetic field. An application of symplectic geometry, from: "Contact and symplectic geometry (Cambridge, 1994)", Publ. Newton Inst. 8, Cambridge Univ. Press (1996) 131 MR1432462
23 V L Ginzburg, An embedding $S^{2n-1}{\rightarrow}\mathbb{R}^{2n}$, $2n-1{\geq}7$, whose Hamiltonian flow has no periodic trajectories, Internat. Math. Res. Notices (1995) 83 MR1317645
24 V L Ginzburg, A smooth counterexample to the Hamiltonian Seifert conjecture in $\bold R^6$, Internat. Math. Res. Notices (1997) 641 MR1459629
25 V L Ginzburg, B Z Gürel, A $C^2$–smooth counterexample to the Hamiltonian Seifert conjecture in $\mathbb R^4$, Ann. of Math. $(2)$ 158 (2003) 953 MR2031857
26 V L Ginzburg, B Z Gürel, Relative Hofer–Zehnder capacity and periodic orbits in twisted cotangent bundles, Duke Math. J. 123 (2004) 1 MR2060021
27 V L Ginzburg, B Z Gürel, E Kerman, Branching Floer homology, in progress
28 V L Ginzburg, E Kerman, Periodic orbits in magnetic fields in dimensions greater than two, from: "Geometry and topology in dynamics (Winston–Salem, NC, 1998/San Antonio, TX, 1999)", Contemp. Math. 246, Amer. Math. Soc. (1999) 113 MR1732375
29 V L Ginzburg, E Kerman, Periodic orbits of Hamiltonian flows near symplectic extrema, Pacific J. Math. 206 (2002) 69 MR1924819
30 A Gray, Tubes, Addison-Wesley Publishing Company Advanced Book Program (1990) MR1044996
31 M R Herman, Examples of compact hypersurfaces in $\mathbb{R}^{2p}, 2p{\geq}6$, with no periodic orbits, from: "Hamiltonian systems with three or more degrees of freedom (S'Agaró, 1995)", NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 533, Kluwer Acad. Publ. (1999) 126 MR1720888
32 H Hofer, On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990) 25 MR1059642
33 H Hofer, Estimates for the energy of a symplectic map, Comment. Math. Helv. 68 (1993) 48 MR1201201
34 H Hofer, D A Salamon, Floer homology and Novikov rings, from: "The Floer memorial volume", Progr. Math. 133, Birkhäuser (1995) 483 MR1362838
35 H Hofer, C Viterbo, The Weinstein conjecture in cotangent bundles and related results, Ann. Scuola Norm. Sup. Pisa Cl. Sci. $(4)$ 15 (1988) MR1015801
36 H Hofer, C Viterbo, The Weinstein conjecture in the presence of holomorphic spheres, Comm. Pure Appl. Math. 45 (1992) 583 MR1162367
37 H Hofer, E Zehnder, Periodic solutions on hypersurfaces and a result by C. Viterbo, Invent. Math. 90 (1987) 1 MR906578
38 H Hofer, E Zehnder, A new capacity for symplectic manifolds, from: "Analysis, et cetera", Academic Press (1990) 405 MR1039354
39 H Hofer, E Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag (1994) MR1306732
40 M Hutchings, Floer homology of families I, Algebr. Geom. Topol. 8 (2008) 435 MR2443235
41 E Kerman, Periodic orbits of Hamiltonian flows near symplectic critical submanifolds, Internat. Math. Res. Notices (1999) 953 MR1717637
42 E Kerman, New smooth counterexamples to the Hamiltonian Seifert conjecture, J. Symplectic Geom. 1 (2002) 253 MR1959583
43 E Kerman, F Lalonde, Length minimizing Hamiltonian paths for symplectically aspherical manifolds, Ann. Inst. Fourier (Grenoble) 53 (2003) 1503 MR2032941
44 F Lalonde, D McDuff, Hofer's $L^\infty$–geometry: energy and stability of Hamiltonian flows I, II, Invent. Math. 122 (1995) 1, 35 MR1354953
45 F Laudenbach, Homotopie régulière inactive et engouffrement symplectique, Ann. Inst. Fourier (Grenoble) 36 (1986) 93 MR850746
46 M Levi, On a problem by Arnold on periodic motions in magnetic fields, Comm. Pure Appl. Math. 56 (2003) 1165 MR1989230
47 G Liu, G Tian, Weinstein conjecture and GW-invariants, Commun. Contemp. Math. 2 (2000) 405 MR1806943
48 G Lu, The Weinstein conjecture on some symplectic manifolds containing the holomorphic spheres, Kyushu J. Math. 52 (1998) 331 MR1645455
49 G Lu, Gromov–Witten invariants and pseudo symplectic capacities, Israel J. Math. 156 (2006) 1 MR2282367
50 L Macarini, Hofer–Zehnder capacity and Hamiltonian circle actions, Commun. Contemp. Math. 6 (2004) 913 MR2112475
51 L Macarini, F Schlenk, A refinement of the Hofer–Zehnder theorem on the existence of closed characteristics near a hypersurface, Bull. London Math. Soc. 37 (2005) 297 MR2119029
52 D McDuff, Geometric variants of the Hofer norm, J. Symplectic Geom. 1 (2002) 197 MR1959582
53 D McDuff, J Slimowitz, Hofer–Zehnder capacity and length minimizing Hamiltonian paths, Geom. Topol. 5 (2001) 799 MR1871405
54 Y G Oh, Chain level Floer theory and Hofer's geometry of the Hamiltonian diffeomorphism group, Asian J. Math. 6 (2002) 579 MR1958084
55 Y Ostrover, A comparison of Hofer's metrics on Hamiltonian diffeomorphisms and Lagrangian submanifolds, Commun. Contemp. Math. 5 (2003) 803 MR2017719
56 L Polterovich, Geometry on the group of Hamiltonian diffeomorphisms, from: "Proceedings of the International Congress of Mathematicians, Vol II (Berlin, 1998)" (1998) 401 MR1648090
57 L Polterovich, The geometry of the group of symplectic diffeomorphisms, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag (2001) MR1826128
58 M Poźniak, Floer homology, Novikov rings and clean intersections, from: "Northern California Symplectic Geometry Seminar", Amer. Math. Soc. Transl. Ser. 2 196, Amer. Math. Soc. (1999) 119 MR1736217
59 F Schlenk, Applications of Hofer's geometry to Hamiltonian dynamics, Comment. Math. Helv. 81 (2006) 105 MR2208800
60 M Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math. 193 (2000) 419 MR1755825
61 D Salamon, Lectures on Floer homology, from: "Symplectic geometry and topology (Park City, UT, 1997)", IAS/Park City Math. Ser. 7, Amer. Math. Soc. (1999) 143 MR1702944
62 K F Siburg, New minimal geodesics in the group of symplectic diffeomorphisms, Calc. Var. Partial Differential Equations 3 (1995) 299 MR1385290
63 J C Sikorav, Systèmes Hamiltoniens et topologie symplectique, ETS, EDITRICE Pisa (1990)
64 M Struwe, Existence of periodic solutions of Hamiltonian systems on almost every energy surface, Bol. Soc. Brasil. Mat. $($N.S.$)$ 20 (1990) 49 MR1143173
65 I Ustilovsky, Conjugate points on geodesics of Hofer's metric, Differential Geom. Appl. 6 (1996) 327 MR1422339
66 H Weyl, On the Volume of Tubes, Amer. J. Math. 61 (1939) 461 MR1507388