## Geometry & Topology

### Action and index spectra and periodic orbits in Hamiltonian dynamics

#### Abstract

The paper focuses on the connection between the existence of infinitely many periodic orbits for a Hamiltonian system and the behavior of its action or index spectrum under iterations. We use the action and index spectra to show that any Hamiltonian diffeomorphism of a closed, rational manifold with zero first Chern class has infinitely many periodic orbits and that, for a general rational manifold, the number of geometrically distinct periodic orbits is bounded from below by the ratio of the minimal Chern number and half of the dimension. These generalizations of the Conley conjecture follow from another result proved here asserting that a Hamiltonian diffeomorphism with a symplectically degenerate maximum on a closed rational manifold has infinitely many periodic orbits.

We also show that for a broad class of manifolds and/or Hamiltonian diffeomorphisms the minimal action-index gap remains bounded for some infinite sequence of iterations and, as a consequence, whenever a Hamiltonian diffeomorphism has finitely many periodic orbits, the actions and mean indices of these orbits must satisfy a certain relation. Furthermore, for Hamiltonian diffeomorphisms of $ℂℙn$ with exactly $n+1$ periodic orbits a stronger result holds. Namely, for such a Hamiltonian diffeomorphism, the difference of the action and the mean index on a periodic orbit is independent of the orbit, provided that the symplectic structure on $ℂℙn$ is normalized to be in the same cohomology class as the first Chern class.

#### Article information

Source
Geom. Topol., Volume 13, Number 5 (2009), 2745-2805.

Dates
Accepted: 29 June 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800322

Digital Object Identifier
doi:10.2140/gt.2009.13.2745

Mathematical Reviews number (MathSciNet)
MR2529945

Zentralblatt MATH identifier
1172.53052

Subjects
Primary: 53D40: Floer homology and cohomology, symplectic aspects
Secondary: 37J10: Symplectic mappings, fixed points

#### Citation

Ginzburg, Viktor L; Gürel, Başak Z. Action and index spectra and periodic orbits in Hamiltonian dynamics. Geom. Topol. 13 (2009), no. 5, 2745--2805. doi:10.2140/gt.2009.13.2745. https://projecteuclid.org/euclid.gt/1513800322

#### References

• P Albers, A note on local Floer homology
• M Audin, J Lafontaine (editors), Holomorphic curves in symplectic geometry, Progress in Math. 117, Birkhäuser Verlag, Basel (1994)
• J-F Barraud, O Cornea, Lagrangian intersections and the Serre spectral sequence, Ann. of Math. $(2)$ 166 (2007) 657–722
• P Biran, L Polterovich, D Salamon, Propagation in Hamiltonian dynamics and relative symplectic homology, Duke Math. J. 119 (2003) 65–118
• Y V Chekanov, Hofer's symplectic energy and Lagrangian intersections, from: “Contact and symplectic geometry (Cambridge, 1994)”, (C B Thomas, editor), Publ. Newton Inst. 8, Cambridge Univ. Press (1996) 296–306
• K Cieliebak, V L Ginzburg, E Kerman, Symplectic homology and periodic orbits near symplectic submanifolds, Comment. Math. Helv. 79 (2004) 554–581
• C C Conley, Lecture, University of Wisconsin (1984)
• M Entov, K-area, Hofer metric and geometry of conjugacy classes in Lie groups, Invent. Math. 146 (2001) 93–141
• M Entov, L Polterovich, Rigid subsets of symplectic manifolds
• M Entov, L Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not. (2003) 1635–1676
• A Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988) 513–547
• A Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 (1988) 775–813
• A Floer, Cuplength estimates on Lagrangian intersections, Comm. Pure Appl. Math. 42 (1989) 335–356
• A Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989) 575–611
• A Floer, Witten's complex and infinite-dimensional Morse theory, J. Differential Geom. 30 (1989) 207–221
• B Fortune, A symplectic fixed point theorem for ${\bf C}{\rm P}\sp n$, Invent. Math. 81 (1985) 29–46
• B Fortune, A Weinstein, A symplectic fixed point theorem for complex projective spaces, Bull. Amer. Math. Soc. $($N.S.$)$ 12 (1985) 128–130
• J Franks, M Handel, Periodic points of Hamiltonian surface diffeomorphisms, Geom. Topol. 7 (2003) 713–756
• U Frauenfelder, F Schlenk, Hamiltonian dynamics on convex symplectic manifolds, Israel J. Math. 159 (2007) 1–56
• V L Ginzburg, The Conley conjecture
• V L Ginzburg, The Weinstein conjecture and theorems of nearby and almost existence, from: “The breadth of symplectic and Poisson geometry”, (J E Marsden, T S Ratiu, editors), Progr. Math. 232, Birkhäuser, Boston (2005) 139–172
• V L Ginzburg, Coisotropic intersections, Duke Math. J. 140 (2007) 111–163
• V L Ginzburg, B Z Gürel, Generic existence of infinitely many periodic orbits in Hamiltonian dynamics, in preparation
• V L Ginzburg, B Z Gürel, Local Floer homology and the action gap
• V L Ginzburg, B Z Gürel, Periodic orbits of twisted geodesic flows and the Weinstein–Moser theorem, to appear in Comment. Math. Helv.
• V L Ginzburg, B Z Gürel, Relative Hofer–Zehnder capacity and periodic orbits in twisted cotangent bundles, Duke Math. J. 123 (2004) 1–47
• V L Ginzburg, E Kerman, Homological resonances for Hamiltonian diffeomorphisms and Reeb flows, Int. Math. Res. Not. (2009) Art. ID rnp 120, 16 pp
• V Guillemin, V L Ginzburg, Y Karshon, Moment maps, cobordisms, and Hamiltonian group actions, Math. Surveys and Monogr. 98, Amer. Math. Soc. (2002) Appendix J by M Braverman
• B Z Gürel, Totally non-coisotropic displacement and its applications to Hamiltonian dynamics, Commun. Contemp. Math. 10 (2008) 1103–1128
• N Hingston, Subharmonic solutions of Hamiltonian equations on tori, to appear in Ann. of Math. Available at \setbox0\makeatletter\@url http://comet.lehman.cuny.edu/sormani/others/hingston.html {\unhbox0
• H Hofer, Lusternik–Schnirelman-theory for Lagrangian intersections, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988) 465–499
• H Hofer, D A Salamon, Floer homology and Novikov rings, from: “The Floer memorial volume”, (H Hofer, C H Taubes, A Weinstein, E Zehnder, editors), Progr. Math. 133, Birkhäuser, Basel (1995) 483–524
• H Hofer, C Viterbo, The Weinstein conjecture in the presence of holomorphic spheres, Comm. Pure Appl. Math. 45 (1992) 583–622
• H Hofer, E Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel (1994)
• E Kerman, Squeezing in Floer theory and refined Hofer–Zehnder capacities of sets near symplectic submanifolds, Geom. Topol. 9 (2005) 1775–1834
• E Kerman, Hofer's geometry and Floer theory under the quantum limit, Int. Math. Res. Not. (2008) Art. ID rnm 137, 36pp
• H V Lê, K Ono, Cup-length estimates for symplectic fixed points, from: “Contact and symplectic geometry (Cambridge, 1994)”, (C B Thomas, editor), Publ. Newton Inst. 8, Cambridge Univ. Press (1996) 268–295
• G Liu, G Tian, Weinstein conjecture and GW–invariants, Commun. Contemp. Math. 2 (2000) 405–459
• G Lu, Gromov–Witten invariants and pseudo symplectic capacities, Israel J. Math. 156 (2006) 1–63
• D McDuff, Hamiltonian $S\sp 1$–manifolds are uniruled, Duke Math. J. 146 (2009) 449–507
• D McDuff, D Salamon, Introduction to symplectic topology, Oxford Math. Monogr., Oxford Science Publ., The Clarendon Press, Oxford Univ. Press, New York (1995)
• D McDuff, D Salamon, $J$–holomorphic curves and symplectic topology, Amer. Math. Soc. Coll. Publ. 52, Amer. Math. Soc. (2004)
• Y-G Oh, A symplectic fixed point theorem on $T\sp {2n}\times{\bf C}{\rm P}\sp k$, Math. Z. 203 (1990) 535–552
• Y-G Oh, Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds, from: “The breadth of symplectic and Poisson geometry”, (J E Marsden, T S Ratiu, editors), Progr. Math. 232, Birkhäuser, Boston (2005) 525–570
• K Ono, On the Arnold conjecture for weakly monotone symplectic manifolds, Invent. Math. 119 (1995) 519–537
• Y Ostrover, Calabi quasi-morphisms for some non-monotone symplectic manifolds, Algebr. Geom. Topol. 6 (2006) 405–434
• S Piunikhin, D Salamon, M Schwarz, Symplectic Floer-Donaldson theory and quantum cohomology, from: “Contact and symplectic geometry (Cambridge, 1994)”, (C B Thomas, editor), Publ. Newton Inst. 8, Cambridge Univ. Press (1996) 171–200
• M Poźniak, Floer homology, Novikov rings and clean intersections, from: “Northern California Symplectic Geometry Seminar”, (Y Eliashberg, D Fuchs, T Ratiu, A Weinstein, editors), Amer. Math. Soc. Transl. Ser. 2 196, Amer. Math. Soc. (1999) 119–181
• D A Salamon, Lectures on Floer homology, from: “Symplectic geometry and topology”, (Y Eliashberg, L Traynor, editors), IAS/Park City Math. Series 7, Amer. Math. Soc. (1999) xiv+430
• D A Salamon, E Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45 (1992) 1303–1360
• M Schwarz, A quantum cup-length estimate for symplectic fixed points, Invent. Math. 133 (1998) 353–397
• M Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math. 193 (2000) 419–461
• A Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Math. 1764, Springer, Berlin (2001)
• M Usher, Spectral numbers in Floer theories, Compos. Math. 144 (2008) 1581–1592
• M Usher, Floer homology in disk bundles and symplectically twisted geodesic flows, J. Mod. Dyn. 3 (2009) 61–101
• C Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992) 685–710
• C Viterbo, Functors and computations in Floer homology with applications. I, Geom. Funct. Anal. 9 (1999) 985–1033